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The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice

We present a detailed study of the geometric and algebraic properties of the multidimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar; the discrete analogue of a conjugate net) and of its basic reductions. To make this study, we introduce the notions of forward and backward data, which allow us to give a geometric meaning to the tau-function of the lattice, defined as the potential connecting these data. Together with the known circular lattice (a lattice whose elementary quadrilaterals can be inscribed in circles; the discrete analogue of an orthogonal conjugate net) we introduce and study two other basic reductions of the quadrilateral lattice: the symmetric lattice, for which the forward and backward data coincide, and the D-invariant lattice, characterized by the invariance of a certain natural frame along the main diagonal. We finally discuss the Egorov lattice, which is, at the same time, symmetric, circular and D-invariant. The integrability properties of all these lattices are established using geometric, algebraic and analytic means; in particular we present a D-bar formalism to construct large classes of such lattices. We also discuss quadrilateral hyperplane lattices and the interplay between quadrilateral point and hyperplane lattices in all the above reductions.


Introduction
In a recent paper [15] we have introduced the notion of "Multidimensional Quadrilateral Lattice" (MQL), i.e., a lattice x : Z N → P M , N ≤ M, with all its elementary quadrilaterals planar, which is the discrete analogue of a multidimensional conjugate net [8]. Furthermore, we showed that the planarity constraint (which is a linear constraint) provides a way to construct the lattice uniquely, once a suitable set of initial data is given.
In this paper we present a detailed study of three basic and independent integrable reductions of the quadrilateral lattice: the symmetric lattice, the circular lattice and the d-invariant lattice; we also study the Egorov lattice which is, at the same time, symmetric, circular and d-invariant. All these reductions satisfy additional geometric properties which are compatible with the planarity constraint of the MQL.
The symmetric lattice follows from the observation that one can associate, with a given quadrilateral lattice, forward and backward data connected through a potential coinciding with the τ -function of the lattice, and it corresponds to the particular situation in which the backward and forward rotation coefficients coincide. The circular lattice, discrete analogue of an orthogonal net, is instead characterized by the fact that all its elementary quadrilaterals are inscribed in circles. The d-invariant lattice is a MQL characterized by the invariance of a certain natural frame along the main diagonal. The Egorov lattice, discrete analogue of a Egorov net [8,1], is simultaneously symmetric, circular and d-invariant (for N = M), and can be equivalently characterized by the fact that a pair of opposite angles of the elementary quadrilateral consists of right angles.
The geometric properties characterizing the above reductions make use of the connections between point lattices and hyperplane lattices (lattices in the dual space (P M ) * ). In some cases the connection comes from additional structure in the ambient space P M ; in some other cases, it is a consequence of the inner symmetry of the lattice. The precise connections between point and hyperplane lattices corresponding to all the above reductions are also presented in this paper.
Our presentation reflects the effort of constructing a general theory of the MQL and of its reductions and therefore the results will not appear in a chronological order of derivation but rather in a logical order.
Although the research field of integrable discrete geometry is relatively new, the amount of associated results is already very large and it is often difficult to go through the corresponding literature, also because many of these results are not even published, having being presented only during conferences or seminars, or private conversations. A brief but hopefuly correct account of the literature closed to the subject considered in this paper is the following.
The proper discrete analogue of a conjugate net on a surface was first proposed by Sauer [34]. The MQL equations were first derived by Bogdanov and Konopelchenko [5] as integrable discrete analogues of the Darboux equations for conjugate nets, but with-out any geometric characterization. The notion of circular lattice was first proposed by Martin, de Pont and Scharrock [30] and Nutborne [32] for N = 2, M = 3, as a discrete analogue of surfaces parametrized by curvature lines (see also [3]); later by Bobenko for N = M = 3 [2] and, finally, for arbitrary N ≤ M by Cieśliński, Doliwa and Santini [7]; subsequently Konopelchenko and Schief have shown [26] that circular lattices in E 3 can be conveniently characterized by solutions of the 2 + 1 dimensional discrete sine-Gordon equation [31]. A geometric proof of the integrability of the circular lattice was first given in [7] while the analytic proof of its integrability was given in [16] through the∂ method. The notion of Egorov lattice with its right angles characterization was found by Schief [35]. In the derivation of the Egorov lattice, he apparently used the algebraic formulation of the symmetric constraint; this formulation was restricted to the subclass of circular lattices and its geometric meaning was not given [36]. He also found the d-invariance of the Egorov lattice (the Killing vector property) [37]. The finite-gap formulations of the circular and Egorov lattices have also recently appeared in the literature [27].
The new results written down in this paper, although already presented in several occasions [10], are the following: 1) The geometric meaning of the τ -function of the MQL; 2) The theory of integrable hyperplane lattices, and its central role in the reduction theory of MQL; 3) The algebraic and geometric notions of symmetric and d-invariant lattices, as basic and independent reductions of the MQL; 4) The successful application of the∂ reduction method, already used in the case of circular lattices [16], to all the other reductions.
After this work was completed and presented we were told that, according to Schief [38], the algebraic definition of a symmetric lattice was presented in [35].
In the rest of this Introduction we summarize the basic results on quadrilateral lattices and the known facts on hyperplanes in projective spaces which will be used in the paper.
In Section 2 we introduce the "backward" representation of the quadrilateral lattice and we show that the compatibility between the backward construction and the standard forward construction leads to the existence of a potential which can be identified with the τ -function of the lattice.
In Section 3 we first introduce the notion of quadrilateral hyperplane lattice; then we introduce and study the notions of dual, adjoint, conjugate and complementary systems of point and hyperplane lattices.
In Section 4 we study the first integrable reduction, the symmetric lattice together with its integrability properties.
In Section 5 we discuss, in the same spirit, the second basic reduction, the circular lattice.
In Section 6 we define the third basic reduction, the d-invariant lattice and study its properties. Section 7 is devoted to the study of the Egorov lattice which is, at the same time, symmetric, circular and d-invariant.
In Section 8 we finally study the integrability properties of all the above lattices from the point of view of their solvability, making use of a∂-reduction method recently introduced in [43] in the continuous case and generalized in [16] to a discrete context.
We finally remark that the equations characterizing the above lattices are potentially relevant also in physics, being integrable discretizations of equations arising in hydrodynamics [19,25,22,39] and in quantum field theory [20,18,9].

Quadrilateral point lattices
Consider a multidimensional quadrilateral lattice; i. e., a mapping x : Z N → P M , N ≤ M, with all the elementary quadrilaterals planar [15]. In the affine representation (in which the lattice is a mapping x : Z N → R M ) the planarity condition can be formulated in terms of the Laplace equations where T i is the translation operator in the i direction, ∆ i = T i − 1 and the coefficients A ij satisfy the MQL equation It is often convenient to reformulate equations (1.1) as a first order system [15]. We introduce the suitably scaled tangent vectors X i , i = 1, ..., N, in such a way that the j-th variation of X i is proportional to X j only (see Figure 1) The compatibility condition for the system (1.4) gives the following new form of the MQL equations The scaling factors H i , called the Lamé coefficients, solve the linear equations whose compatibility gives equations (1.5) again; moreover In [15] it was proven that, given N(N − 1)/2 initial quadrilateral surfaces, the quadrilateral lattice x follows uniquely from the planarity constraint. To construct the initial surfaces, one gives N arbitrary intersecting initial curves x (0) i , i = 1, . . . , N; the initial (ij)-surface is then built uniquely assigning the initial data A (0) ij , i = j, as functions of n i , n j , via equation (1.1). Equivalently, together with the N intersecting initial curves, we can give the initial data {H ij }, meaning that we give the coefficients H ij , i = j, as functions of n i , n j . Therefore the solution of the MQL equations depends on N(N − 1) arbitrary functions of two variables. Remark. To make the construction of the lattice possible, in our considerations we assume we deal with generic lattices, i.e., that the point x and its nearest neighbours T 1 x, . . . , T N x are in general position; in consequence, the subspace x, T 1 x, . . . , T N x is a linear subspace of P M of maximal possible dimension N.
In this paper we study some distinguished reductions of the MQL which posses additional geometric properties that, once imposed on the initial surfaces "propagate" everywhere through the construction of the lattice. Since the quadrilateral lattice is integrable, these reductions will inherit its integrability properties.
In the continuous limit: (1.10)

Hyperplanes
In Section 3 we introduce and study the properties of lattices in the dual space, i.e., of hyperplane lattices. These considerations will turn out to be relevant in the reduction theory of the quadrilateral lattices, when the introduction of additional geometric structure will allow to establish a direct connection between point lattices and hyperplane lattices.
To make the paper self contained, in the rest of this introduction we summarize some basic known facts on the algebraic representation of the projective space and of its dual.
Points of P M are directions (one dimensional linear subspaces) of R M +1 and they can be represented (up to multiplication by a non-zero scalar factor) by nonzero vectors of R M +1 . In a fixed basis e 0 , e 1 , . . . , e M of R M +1 , the coordinates u = (u 0 , u 1 , . . . u M ) T of such a vector are called the homogeneous coordinates of the corresponding point u = [u] of the projective space.
The hyperplanes of P M are M dimensional linear subspaces of R M +1 and they can be represented (up to multiplication by a non-zero scalar factor) by non-zero covectors of (R M +1 ) * ≡ R M +1 . The coordinates a * = (a * 0 , a * 1 , . . . a * M ) of such a co-vector are called the homogeneous coordinates of the corresponding hyperplane a * = [a * ], and the condition that the point with homogeneous coordinates u = (u 0 , u 1 , . . . , u M ) T belongs to the hyperplane represented by a * = (a * 0 , a * 1 , . . . , a * M ) is given by the linear homogeneous equation Remark (Duality principle). Notice that equation (1.11) is "symmetric" in the sense that the expression "the point u belongs to the hyperplane a * " can be changed into "the hyperplane a * contains the point u". Geometrically, all hyperplanes (points of (P M ) * ) passing through a fixed point of P M form a hyperplane in (P M ) * , which is represented by this point; therefore ((P M ) * ) * = P M .
By fixing a hyperplane P M −1 ∞ in P M , called then the hyperplane at infinity, we can represent the remaining (affine) part A M = P M \ P M −1 ∞ of the projective space by points v ∈ R M ; if the hyperplane at infinity is characterized by u 0 = 0, then the points of the affine space can be normalized to (1, u 1 , . . . , u M ) T , and u = (u 1 , . . . , u M ) T .
Hyperplanes in A M can be represented (again, up to a non-zero factor) by nonhomogeneous linear equations as follows The representation can be made unique, by affinization of (P M ) * , i.e., by removing from (P M ) * hyperplanes passing through a fixed point of P M . For our purposes we assume that this point belongs to A M , and we identify it with the origin of R M . Then the equation of any hyperplane which does not pass through the origin, i.e., a * 0 = 0, can be normalized to have a * 0 = −1. Such a hyperplane can be represented by the co-vector a * ∈ (R M ) * and consists of points x satisfying the equation If v is a point of the hyperplane represented by a * , then the parallel (in the standard sense) hyperplane passing through t v is represented by t −1 a * ; equivalently, the equation of such a hyperplane can be written as a * | x = t. Taking the limit t → ∞ we infer that the hyperplane at infinity P M −1 ∞ is represented by the zero co-vector 0 * . On the other hand, all the hyperplanes passing through 0 ∈ R M are represented by "infinite" co-vectors; equivalently, the equation of the hyperplane passing through 0 and parallel to that represented by a * can be written as a * | x = 0.
Given two hyperplanes a * and b * represented by the co-vectors a * and b * , the equation of the unique hyperplane passing through the origin and containing their intersection a * ∩ b * is (1.14) Definition 1.1. Two subspaces of co-dimension 2 are called "co-parallel" if there exists a hyperplane passing through 0 and containing them.
Remark. The above notion is dual to the parallelism of two lines in the affine space.
Corollary 1.1. Two co-dimension 2 subspaces obtained by intersection of two pairs of hyperplanes a * i ∩b * i , i = 1, 2, are co-parallel if the corresponding co-vectors a * i − b * i , i = 1, 2 are proportional.
A correlation is a projective mapping between a projective space and its dual In the homogeneous description, such a mapping is given by a linear mapping (given uniquely up to a non-zero scalar factor) between the vector space R M +1 and its dual; if a * = [a * ], v = [v], and a * = C(v), then the correlation C is represented by a matrix C such that a * = (Cv) T .
Any correlation C defines its adjoint correlation Any polarization P defines the corresponding quadric hypersurface Q P , which consists of points belonging to their polar hyperplanes: x ∈ P(x); in the homogeneous description, the quadric is given by equation P x|x = 0. Consider the polarization whose quadric is the standard sphere of radius 1 centered at the origin: Then the polar hyperplane of a point v is the hyperplane orthogonal to v and passing through the point v/( v · v) (see Figure 2). The polar of the origin is the hyperplane at infinity, therefore this polarization is an affine mapping, i.e., it maps parallel lines into co-parallel subspaces (of co-dimension two).

The backward representation of the quadrilateral lattice
In this Section we define the backward dataX i ,H i ,Q ij of the quadrilateral lattice. It turns out that the relation between the standard forward data X i , H i , Q ij and the backward data is given in terms of the τ -function, which is one of central objects of the soliton theory. The backward tangent vectorsX i and the backward Lamé coefficientsH i , i = 1, . . . , N are defined with the help of the backward difference operator∆ i := 1 − T −1 i : the backward Lamé coefficients are again chosen in such a way (see Figure 3) that the∆ i variation ofX j is proportional toX i only. We define the backward rotation coefficientsQ ij as the corresponding proportionality factors

4)
where, in the notation of this Section, The forward and backward rotation coefficients Q ij andQ ij describe the same lattice x from different points of view, therefore one can expect their interrelation. Indeed, defining the functions ρ i : Z N → R as the proportionality factors between X i and T iX i (both vectors are proportional to ∆ i x): we have the following Proposition 2.2. The forward and backward data of the lattice x are related through the following formulas and the factors ρ i are first potentials satisfying equations Proof. Using equations (2.6), (2.2) and (1.4) we obtain which, by comparing coefficients in front of the vectors X i , X j , leads to equations (2.7) and (2.8).
Remark. Since Q ij andQ ij are both solutions of the MQL equations (1.5), then equations (2.6)-(2.8) describe a special symmetry transformation of equations (1.5), first found in [26] without any associated geometric meaning. The RHS of equation (2.8) is symmetric with respect to the interchange of i and j, which implies the existence of a potential τ : Z N → R, such that therefore equation (2.8) defines the second potential τ : The potential τ connecting the forward and backward data: is the famous τ -function of the quadrilateral lattice. 14) then equation (2.8) can be rewritten as 15) and the MQL equations (1.5) take the form Remark. The τ -function representation of the MQL equations was found in [18] using the Miwa transformation of the τ -function representation of the Darboux equations.
We notice that, for a given lattice x, the forward data {X i , Q ij } are defined up to rescaling by functions a i (n i ) expressing the freedom in the definition of the vectors X (0) i on the initial curves. An analogous freedom exists for the backward data The corresponding rescaling of ρ i and τ is therefore given by Finally, we remark that the product (T i Q ji )(T j Q ij ), which appears in the definition of the τ -function, is the ratio of the areas of the two affine parallelograms P (∆ i X j , ∆ j X i ) and P (X i , X j ) (see Figure 4).
Unlike the definitions of the forward and backward rotation coefficients, this product is invariant with respect to their possible redefinitions given by equation (2.17). It can be seen expressing the product, using equations (1.7), in terms of the data A ij as follows .
Observe finally that equation (2.7) leads to which implies that the discussed product quantity is also the ratio of the areas of the backward parallelograms P (

Hyperplane lattices
Consider a lattice y * : Z N → (P M ) * , N ≤ M, in the space of hyperplanes of P M , which we call the hyperplane lattice. The space (P M ) * , called also dual space to P M , has a natural projective structure and a priori one expects that the algebraic description of the quadrilateral lattices in the dual space be the same like that of quadrilateral point lattices and, therefore, the considerations of the previous Sections can be applied to hyperplane lattices without essential modifications. However, this Section is devoted to investigate hyperplane lattices from a geometric point of view and to make clear the geometric content of their algebraic description.

Quadrilateral hyperplane lattices
The basic property of quadrilateral lattices, i.e., the planarity of their elementary quadrilaterals, when applied to hyperplane lattices, can be formulated as follows.
To explain this definition notice that, if the hyperplane lattice is given in homogeneous coordinates by the function y * : Z N → (R M +1 ) * \ {0 * }, then Definition 3.1 states that the four co-vectors T i T j y * , T i y * , T j y * , and y * are linearly dependent. If T i y * , T j y * , y * are linearly independent, then the co-vector T i T j y * representing the hyperplane T i T j y * is a linear combination of the co-vectors y * , T i y * and T j y * T i T j y * = αT i y * + βT j y * + γy * . (3.1) This equation can be transformed into the Laplace equation In the affine gauge, the coefficients α, β and γ of the decomposition (3.1) are subjected to the constraint α + β + γ = 1 (3.3) and the equation (3.2) reduces to Remark. In our considerations we always assume we deal with generic lattices, i.e., that the hyperplane y * and its forward neighbors T 1 y * , . . . , T N y * (and backward neighbors T −1 1 y * , . . . , T −1 N y * ) are in general position, i.e., their equations are linearly independent. In consequence, the intersection y * ∩ T 1 y * ∩ · · · ∩ T N y * (and y * ∩ T −1 Example 3.1. Given a two dimensional quadrilateral lattice x in the three dimensional projective space, define the lattice y * of the hyperplanes passing through x, T 1 x and T 2 x. Because of the planarity of the elementary quadrilaterals of x, it is easy to see that the four hyperplanes y * , T 1 y * , T 2 y * and T 1 T 2 y * intersect in the point T 1 T 2 x. Therefore, the (hyper)plane lattice y * is quadrilateral.
Correlations map quadrilateral point lattices into quadrilateral hyperplane lattices.

Dual systems
We first recall that a quadrilateral lattice x ′ is called parallel to the quadrilateral lattice x [17] (or obtained from x via the Combescure transformation), if the tangents to both lattices are parallel in the corresponding points: In consequence, the scaled tangent vectors X ′ i of the lattice x ′ can be choosen to be equal to those of the lattice x: X ′ i = X i ; then the rotation coefficients of both lattices coincide as well: Q ij = Q ′ ij , and the Lamé coefficients H i and H ′ i are solutions of the same equation. In this Section we will learn how to construct quadrilateral hyperplane lattices using systems of parallel quadrilateral point lattices.
. . , M, whose corresponding vectors are linearly independent. Denote by y * (k) , k = 1, . . . , M, the system of hyperplane lattices uniquely defined by the properties that y * (k) passes through x (k) and is spanned by the vectors x (l) , l = k; i.e., We call such a system of hyperplane lattices the dual system to the system of parallel point lattices x (k) .
The aim of this Section is to prove that the hyperplane lattices y * (k) are quadrilateral hyperplane lattices.
k=1 in the ambient space R M and arrange the parallel system of point lattices in the matrix Ω of the system: equivalently, the matrix Ω represents a linear operator Corollary 3.1. The components of the dual system y * (k) in the basis { e * k } M k=1 are given by the rows of the matrix Ω −1 .
Let us arrange the coefficients H i(k) , i = 1, . . . , N, k = 1, . . . , M, into the rowvectors then X * i , i = 1, . . . , N, form a (co)vector valued solution of the adjoint linear problem (1.6) and the matrix Ω can be found from equations It was shown in [29] that the matrix Ω plays a relevant role in the theory of transformations of quadrilateral lattices. The following Theorem, which contains, as particular cases, all the classical transformations of a quadrilateral lattice [17], was proven in [29].
i and Ω be defined as above; then the following functions ij , and the corresponding potential is given by

10)
where C is a constant operator.
Denote by x * (k) the rows of Ω; then The rows x * (k) of the matrix Ω represent a system of co-parallel quadrilateral hyperplane lattices, which we call the adjoint system to x k .
Proof. Let us rewrite equation (3.7) in a backward form , and therefore (see Proposition 2.1) also the forward Laplace equations. Finally, since∆ i x * (k) ∼∆ i x * (l) , then the corresponding co-dimension 2 subspaces of hyperplane lattices are co-parallel in the sense of Definition 1.1.
Remark. Given the parallel system x (k) , k = 1, . . . , M, the corresponding adjoint system x * (k) is given up to a fixed basis used to define Ω; on the contrary, the dual system of hyperplane lattices y * (k) is given uniquely.
Corollary 3.4. Notice that the forward rotation coefficients of the system x (k) are the backward rotation coefficients of the system x * (k) : Q ij =Q * ij .
Combining the above Lemma with Theorem 3.2 we get the following Theorem 3.5. The hyperplane lattices y * (k) of the dual system to the system of parallel quadrilateral point lattices x (k) are co-parallel quadrilateral hyperplane lattices. Corollary 3.6. Equivalently, the forward rotation coefficients of the hyperplane lattice are backward rotation coefficients of its adjoint point lattice.  Remark. We are interested only in a situation in which x is a quadrilateral point lattice and y * is a quadrilateral hyperplane lattice.

The adjoint and conjugate lattices
The notion of conjugacy between point lattices and hyperplane lattices is the natural generalization of the notion of conjugacy between point lattices and rectilinear congruences (line lattices with any two neighbouring lines coplanar) introduced in [17].
As it was also shown in [17], the lattices parallel to x describe transversal congruences conjugate to the quadrilateral point lattice x. Moreover, the tangent congruences can be also obtained in this way via singular limits.
Corollary 3.8. Given a quadrilateral point lattice x in P M and given (M −1) linearly independent congruences conjugate to x, then the hyperplane lattice y * conjugate to x and spanned by the lines of these congruences is quadrilateral.

The complementary lattice
The linear system (1.4) describes the variation of the normalized tangent vectors X i of a quadrilateral point lattice in directions j = i, and leads to the MQL equations (1.5). In this Section we study the variation of the vectors X i in the corresponding i-th directions of the lattice. Discussion of such variations naturally leads to the definition of a hyperplane lattice, which will be called the complementary lattice.
Consider the quadrilateral lattice x : Z N → R M with the given set of tangent vectors X i , i = 1, . . . , N, and the corresponding set of the Lamé and rotation coef-  (3.14) and let us define M − N vectors X a , a = N + 1, . . . , M, via an analogue of equa- Remark. The vectors X a , a = N + 1, . . . , M, are the Combescure transforms of the lattice x, but it was not accidental that we gave to equations (3.14) and (3.15) the form of the Darboux equations (1.5) and of the linear problem (1.4).
When the full set of vectors X k , k = 1, . . . , M, is linearly independent, we obtain, in each point of the lattice x, a basis of the whole space R M ; this type of basis along a quadrilateral lattice has been considered already in [16] and can be called the extended basis along the lattice. ByỸ * We will study properties of the coefficientsP * i ,P * ij and their relation to previously introduced objects.
Using equation (3.16) we obtain that  The compatibility condition of these equations gives the Darboux equations for the backward rotation coefficientsP * ij , i = j = 1, . . . , N gives

26)
where Q ii (and similarlyP * ii ) are potentials defined in [16] for any solution of the MQL system, by the equations Moreover, from the same compatibility, we obtain the following equation To make the above considerations symmetric we define a hyperplane lattice which has the vectorsỸ * i , i = 1, . . . , N, as normalized backward tangent vectors, andP * ij , i = j = 1, . . . , N as backward rotation coefficients. Definition 3.6. Given the quadrilateral lattice x : Z N → R M together with its extended frame X k and its dualỸ k , k = 1, . . . , M, define the complementary lattice of x as via solution of the following compatible equations whereF * i , i = 1, . . . , N, is a solution of the system (3.25), interpreted now as the adjoint of the linear system (3.22) Remark. The additional vectorsỸ * a and functionsP * ia , a = N + 1, . . . , M play a role similar to that of X a and Q ai .
By simple calculation one can obtain the following result.
Finally, we present a Theorem, which can be proved by simple algebra using formulas of Corollaries 3.10 and 3.11, and which contains a geometric characterization of the complementary lattice.

The Symmetric Lattice
Definition 4.1. A quadrilateral lattice x is symmetric iff its forward rotation coefficients are its backward rotation coefficients as well; i.e. (4.1) The considerations of Section 2 imply the following characterization.
Proposition 4.1. A quadrilateral lattice is symmetric iff, for a given set of rotation coefficients Q ij , there exists a τ -function of the lattice such that 2) or equivalently, in terms of the corresponding first potentials ρ i , Remark. Due to equations (2.17)-(2.19), the above definition is independent of the particular choice of the rotation coefficients Q ij .

It turns out that
Proposition 4.2. The symmetric lattice is an integrable reduction of the quadrilateral lattice.
Proof. Recall that, from a geometric point of view, the integrability of a reduction means that, if the reduction condition is satisfied on the initial surfaces, then it must propagate in the construction of the lattice.
As it was shown in [15] the solution Q ij of the MQL equations (1.5) is fixed by the values of the rotation coefficients Q (0) ij on the initial surfaces. Therefore, if Q (0) ij =Q (0) ij on the initial surfaces, then they are equal Q ij =Q ij in the whole lattice, since the backward rotation coefficientsQ ij satisfy the same equations as Q ij .
The algebraic content of this result is instead expressed by the following equation where   Proof. The quadrilateral with the initial vertex is described by the following rotation coefficients: T i Q ji , T j Q ij , T iQji and T jQij connected by equation (2.7). Since then the parallelograms P (T iX i , T jX j ) and P (∆ i X j , ∆ j X i ) are similar (see Figure 5) if and only if which means, due to (2.7), that the backward and forward Q's are equal.

Proposition 4.4.
A quadrilateral lattice is symmetric iff, for a given set of the forward tangent vectors X i of the lattice, there exists a complementary set of the backward tangent vectorsX i such that the parallelograms P (T iX i , T jX j ) and P (∆ i X j , ∆ j X i ) are similar.
Remark. Due to equations (2.17)-(2.19) the above characterization of the symmetric lattice is independent of a particular choice of the vectors X i .
Integrability of the symmetric lattice can be formulated as follows Corollary 4.5. If the system of initial quadrilateral surfaces admits a compatible set of forward-backward data such that P (T iX i , T jX j ) and P (∆ i X j , ∆ j X i ) are similar, then the similarity of the parallelograms holds in the whole quadrilateral lattice.
Remark. Notice that, in order to define the symmetric lattice, we need to know what similar parallelograms are.
The solution of the MQL equations for an N dimensional symmetric lattice depends on N 2 arbitrary functions of two variables, i.e., one half of the arbitrary functions parametrizing the solution of the MQL equations for generic N dimensional quadrilateral lattice (see Section 1.1). Given a symmetric lattice equipped with a compatible set of forward and backward data, denote the similarity factor between the parallelograms P (T iX i , T jX j ) and P (∆ i X j , ∆ j X i ) by σ (ij) = σ (ji) Therefore, to construct the initial (i, j)-surface of a symmetric lattice, one gives two arbitrary intersecting i-and j-curves and, on them, the tangent vectors X j ; one finally gives σ (ij) = σ (ji) as functions of (n i , n j ). The descriptions of the symmetric lattice presented above are not explicit. Indeed they involve statements about the existence of suitable potentials. There exists however another characterization of the symmetric lattice in terms of the forward rotation coefficients only. Theorem 4.6. A quadrilateral lattice is symmetric iff, for different indices i, j, k, its rotation coefficients satisfy the following constraint (4.10) In the proof we will use two simple facts (equations (4.11)-(4.12) and (4.15) below) valid for a generic quadrilateral lattice.
For a given set of the compatibile forward and backward rotation coefficients Q ij andQ ij , define functions R ij as then from equations (2.7) it follows that The MQL equation (1.5) can be written as (4.14) interchanging the indices j and k in the second equation and eliminating T k T j Q ij we obtain Proof of the Theorem. The implication (4.3) ⇒ (4.10) is obvious. Let us concentrate on the opposite implication. Let us start from any set of backward rotation coefficientsQ ij related with Q ij via equations (2.6) (2.7); the condition (4.10) implies which, together with (4.15) and with the corresponding formula satisfied by the backward rotation coefficientsQ ij , gives, for j different from i and k, i.e., R ik is a function of n i and n k only. This, together with condition (4.10) written in terms of R ij , as R ij R jk R ki = 1 (4.18) and with equation (4.12), implies the existence of functions a i (n i ) such that We use the functions a i to redefine the potentials ρ i and obtain new backward rotation coefficientsQ ij satisfying Q ij =Q ij .
The above characterization of the symmetric lattice works only when the dimension of the lattice is greater then two. In the following Proposition we present an analogous criterion for N = 2, which can be useful, for example, to check directly if the initial quadrilateral surfaces are symmetric.

Proposition 4.7. A two dimensional quadrilateral lattice is symmetric iff the function
satisfies equation Proof. The implication from (4.3) to (4.21) is trivial. To prove that the condition (4.21) is sufficient we notice that, in terms of R ij , it can be rewritten as which leads again to Remark. In order to check the symmetry condition for the initial surfaces we use the criterion (4.21) supplemented by (4.10) in the points where the initial surfaces meet.
As we have anticipated, the constraints discussed in this paper allow one to establish a connection between quadrilateral point lattices and their duals, the quadrilateral hyperplane lattices. The following proposition describes this connection in the case of the symmetry constraint.
Proposition 4.8. Given a system of parallel quadrilateral lattices { x (k) } M k=1 and the associated matrix Ω defined with respect to an orthonormal basis { e k } M k=1 , e k · e l = δ kl , then the following properties are equivalent. i) The matrix Ω of the system is symmetric: ii) The polar hyperplane P( x (k) ) of the point lattice x (k) coincides with the hyperplane lattice x * (k) : iii) The lattices x (k) , k = 1, . . . , M are symmetric. Furthermore the associated tangent vectors X i and X * i are related in the following way Proof. i) ⇔ ii) The equivalence of i) and ii) follows immediately from the definitions of the potential matrix Ω and of the polar transformation P. i) ⇒ iii). The application of ∆ i to equation (4.24) gives the equations which imply equations X T i = γ i T i X * i for some proportionality factor functions γ i . The linear problem (1.4) and its adjoint (1.6) satisfied by X i and X * i imply that γ i satisfy equations (2.8) (which allows to identify γ i with ρ i ) and lead to the symmetry condition (4.2). iii) ⇒ i) Following a similar strategy, one can show that which implies (4.24), up to some constant of integration.
Corollary 4.9. A quadrilateral lattice x is symmetric iff it is adjoint to its own polar.
Remark. In the continuous limit (1.9) the symmetric quadrilateral lattice reduces to a symmetric conjugate net, for which the rotation coefficients β ij satisfying the Darboux equations (1.10) are symmetric In fact, one should allow for the less restricitve condition which gives (4.29), after an admissible rescaling of the data. The continuous limit of the criterion (4.10) is equivalent to (4.30).

The Circular Lattice
The discrete analogue of an N -dimensional orthogonal system of coordinates is the circular lattice. An elementary characterization of circular quadrilaterals states that, if a circular quadrilateral is convex, then the sum of its opposite angles is π; when the quadrilateral is skew, then its opposite angles are equal. This leads to a convenient characterization of a circular lattice [16].
or, equivalently, It turns out that [7,16] Proposition 5.2. The circular lattice is an integrable reduction of the quadrilateral lattice.
Proof. The proof consists in showing that the circularity property is an admissible constraint for the quadrilateral lattice; i.e., once imposed on the initial surfaces, it propagates transversally through the lattice. This was shown in [7] using purely geometric means. The algebraic proof is instead based on the following formula which is a direct consequence of equations (1.4)-(1.5). We see that, if the circularity constraint (5.2) is satisfied on the initial surfaces (the RHS of (5.3) is zero), then it propagates transversally through the lattice (the LHS of (5.3) is zero).
Corollary 5.3. 1) The circularity constraint (5.2) implies the following formula [16]: which, compared with equations (2.8)-(2.10), allows to fix, without loss of generality, the backward formulation of the circular lattice in the following way: 2) The circularity constraint (5.2), written in terms of the backward data of the lattice, reads as follows:C Proof. 1) Equation (5.6) is a strightforward consequence of equation (5.2) and has been found in [16].
2) Equation (5.7) follows from the equalities The first equality follows from rewriting C • ij in terms of the backward data; the second equality follows from equations straightforward consequence of (2.2).
Other two convenient characterizations of the circular lattice are contained in the following result found in [26] and explained geometrically in [13]. A distinguished sub-class of circular lattices corresponds to the particular case in which the lattice points x belong to the sphere of radius R: | x| = R. In this case there exists, like for the symmetric reduction, an elegant relation between point lattices and hyperplane lattices.
Proposition 5.5. Given a system of parallel quadrilateral lattices { x (k) } M k=1 and the associated matrix Ω of the system defined with respect to an orthonormal basis { e k } M k=1 , then the following properties are equivalent. i) The matrix Ω/R is orthogonal: ii) The polar hyperplane P( x (k) ) coincides with the dual hyperplane R 2 y * (k) : iii) The quadrilateral lattices x (k) /R, k = 1, .., M form an orthonormal basis: In addition, the associated tangent vectors X i , X * i , i = 1, ..., N, are related by the following formulas with and satisfy the circularity constraint (5.2) and its adjoint Proof. The equivalence between i) and ii) and formula (5.12) is a straightforward consequence of the definitions of Ω, x (k) and y * (k) . Furthermore, the quadrilateral lattice on a sphere is obviously circular, the circles being the intersections of the sphere with the planes of the elementary quadrilaterals [13]. i) ⇒ iii). Applying ∆ i to equation (5.10) leads to which implies that for some γ i . Using equation (3.7) in (5.18) one obtains (5.20) which, together with (5.19) leads to identification of the factors γ i Notice that equations (1.6) imply Application of the shift in j direction to equation (5.20) and using the above identity leads to equations the first of them allows for identification ρ i = 2γ i R 2 , while the second gives the circularity condition. At last, equations (5.13) and (5.15) imply the following relation between the circularity property and its dual: which implies that also equation (5.16) is satisfied. The proof of: iii) ⇒ i) is similar and is left to the reader.
Corollary 5.6. Quadrilateral lattice in a sphere is conjugate to its own polar (with respect to the sphere) hyperplane lattice.
In the continuous limit, equations (5.2) become the orthogonality conditions and the circular lattice reduces to an orthogonal conjugate net.

D-invariant lattice
In this Section we introduce and discuss a basic dimensional reduction of the quadrilateral lattice, the d-invariant lattice, characterized by the invariance of a certain natural frame along the main diagonal of the lattice. To do so, it is convenient to put this reduction in the natural framework of the theory of transformations of the quarilateral lattice, discussed in great detail in [17].
From a quadrilateral lattice x : Z N → R M , one can easily construct a new quadrilateral lattice just translating x in some coordinate direction and combining this translation with a Combescure transformation. If the translation takes place along the main diagonal, one abtains the new quadrilateral latticê where T := N i=1 T i is the total translation along the main diagonal and C(·) is the Combescure transformation [17]. From the above definition it follows that where Q ii was defined in (3.27).
Proof. We first prove by induction the case i ∈ L. For |L| = 1 the statement follows from the linear problem (1.4). When k ∈ L and k = i and the upper part of the formula (6.5) holds then and application of the Darboux equations (1.5) concludes the first part of the proof. Notice that applying the shifts T L and T k in different order we obtain the following generalized Darboux equations To show the lower part of the formula (6.5) let us apply the shift T i to the upper part of it obtaining It remains to prove that for a generic lattice and i ∈ L which can be done, again, by simple induction with the help of equation (6.6).
The quadrilateral latticex is characterized by the following property.
The fixed point of transformation (6.1) (and therefore an integrable reduction of the quadrilateral lattice) is represented by the lattices x which are parallel to their translations T x: x = C(T x) or, equivalantly, for which T X i = X i .
Remark. Equation (6.11) implies that Remark. The d-invariant lattice can be described effectively by N − 1 parameters, since it depends on the differences of the variables n i : X i = X i (n 1 − n 2 , n 2 − n 3 , . . . , n N −1 − n N ) . (6.13) A d-invariant lattice is characterized by the following property.
x Figure 6: Egorov lattice Remark. Formula (6.14) implies that the N dimensional d-invariant lattice is effectively contained in an N dimensional subspace of R M , therefore without loss of generality we can put in this Section N = M.
We present now the characterization of d-invariant lattices in terms of hyperplane lattices.
Theorem 6.5. If the quadrilateral lattice x : Z N → R N is d-invariant then its rotation coefficients Q ij are also the backward rotation coefficients of its complementary latticeP * ij = Q ij , i = j = 1, . . . , N. (6.16) Proof. If x is quadrilateral, then comparison of the formula (6.14) with equations (3.17) and (3.26) proves the statement.
Remark. The Egorov lattice constraint can be written algebraically in the form which implies the circularity condition (5.2).
Corollary 7.2. The line x, T i T j x is a main diagonal of the circle defined by the points: x, T i x and T j x.
Proposition 7.3. The Egorov lattice is an integrable reduction of the quadrilateral lattice.
Proof. Define functions C E by equation and notice the following identity: valid for a generic quadrilateral lattice. In the case of the Egorov lattice we have C E = 0, and equation (7.3) shows that such constraint is admissible.
In the previous sections we introduced two other basic integrable reductions of the quadrilateral lattice: the symmetric and the d-invariant lattices. We will show that the Egorov lattice is symmetric and, for N = M, d-invariant.
Proposition 7.4. The Egorov lattice is symmetric.
Proof. The linear problem (1.4) and the constraint (7.1) imply that which gives Because the Egorov lattice is is circular, then |X i | 2 can be identified with the potentials ρ i , therefore equation (7.4) leads to the symmetry constraint (4.3).
Remark. An equivalent form of the constraint (7.4) was used by Schief in his derivation of the Egorov lattice from the circular lattice [36].
Remark. The symmetry and circularity constraints are not enough to obtain algebraically the Egorov lattice. Indeed, consider a symmetric and circular lattice together with its tangent vectors X i and the corresponding rotation coefficients Q ij . The symmetry condition implies the existence of a τ -function (we call it τ S ) such that the potentials ρ S i = T i τ S /τ S satisfy The circularity condition, in turn, implies existence of a τ -function (we call it τ C ) such that the corresponding potentials ρ C i are given by Equations (2.17)- (2.19) imply that the potentials ρ C i and ρ S i are connected by functions of single variables The Egorov lattice corresponds to the distinguished case in which we have a i ≡ 1, i = 1, . . . , N.
Corollary 7.5. In the circular lattice |T iX i | = 1/|X i |, which implies that the parallelogram P (X i , X j ) is anti-similar to the parallelogram P (T iX i , T jX j ). In the Egorov lattice the parallelogram P (X i , X j ) is also anti-similar to the parallelogram P (∆ i X j , ∆ j X i ).
For N = M the Egorov lattice exhibits the d-invariance property [37].
Proposition 7.6. The Egorov lattice x : Proof. The orthogonality conditions (7.1) imply that where T i X ℓ N ℓ=1,ℓ =i is the linear space spanned by {T i X ℓ } N ℓ=1 , ℓ = i. In addition, the planarity of the lattice implies that these two linear subspaces coincide; therefore X i and T X i , which are orthogonal to the same (N − 1) dimensional linear subspace, must be proportional: Applying T to the linear system (1.4) and using (7.10), we infer that a i = a i (n i ) (= 1 without loss of genericity) and T Q ij = Q ij .
We conclude this section considering the Egorov lattice from the point of view of the parallel system x (k) and of its connections with hyperplane lattices. The results are a straightforward consequence of the Propositions 4.8, 5.5 and of the definition of the Egorov lattice.
Proposition 7.7. Given a system of parallel quadrilateral lattices { x (k) }, k = 1, .., M and the associated matrix Ω, then the following properties are equivalent.

8∂ Formulations of the Reductions
In this section we prove that the distinguished reductions of the quadrilateral lattice discussed in the previous sections are integrable via the∂ reduction method introduced in [43] and generalized to a discrete context in [16]. For the sake of completeness, we first summarize in Sections 8.1 and 8.2 the∂ formulation of the quadrilateral lattice and the main result of the∂ reduction theory applied to it. The∂ dressing method is a very convenient tool to construct integrable multidimensional systems, together with large classes of solutions [42,4,40]. Consider the (by assumption uniquely solvable) matrix M × M∂ problem where ∂λ = ∂/∂λ, the given rational function η(λ) (the normalization of φ(λ)) describes the singularities and the asymptotic behaviour of φ in the complex plane and R(λ, λ ′ ) is the given M × M matrix∂ -datum; consider also the adjoint∂ problem: The above∂ problems imply the bilinear identity: (where C ∞ is the circle with center at the origin and arbitrarily large radius, and the corresponding integration is counter-clockwise), which involves the solutions φ 1 and φ * 2 of (8.1) and (8.2), corresponding to the normalizations η 1 and η 2 respectively. The dependence of the M × M matrices φ, φ * and R onλ andλ ′ : φ = φ(λ,λ), R = R(λ,λ, λ ′ ,λ ′ ) will be omitted systematically throughout the paper.

8.1∂ Formulation of the Quadrilateral Lattice
It turns out that the MQL equations are integrable via the∂ -dressing method [5,16] and all the geometric quantities of the lattice have a distinguished role in this ∂ scheme.

8.2∂-reduction Theory of the Quadrilateral Lattice
The above∂ formulation allows one to look for reductions of the MQL at the simpler level of the∂ -datum R [16]. The particular form (8.4) of it implies the following: The following linear constraint on the∂ datum R(λ, λ ′ ): gives rise to integrable reductions of the MQL. In formula (8.37) and A(λ) is an arbitrary diagonal matrix.
The main implication of the constraint (8.37) is that the function φ T (λ −1 )F (λ) satisfies the adjoint∂ problem (8.2) while the function F −1 (λ −1 )φ * T (λ −1 ) satisfies thē ∂ problem (8.1): (8.40) and these equations, through the bilinear identity (8.3), imply the nonlocal quadratic constraints: Therefore the constraint (8.37) establishes a nontrivial connection, whose nature depends on the particular choice of F (λ) (or, better, of A(λ)), between the solutions of the∂ problem (8.1) and of its adjoint (8.2) or, equivalently, between quadrilateral lattices and their dual objects, the quadrilateral hyperplane lattices. In the following we shall identify the matrix functions A(λ) which correspond to the symmetric, circular and Egorov lattices.