{'fraction_non_alphanumeric': 0.04760136003885825, 'fraction_numerical': 0.028572244921283466, 'mean_word_length': 3.8862208571827446, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 65, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Zeroth-order conservation laws of two-dimensional shallow water equations with variable bottom topography

We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth-order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating the classification of zeroth-order conservation laws of these systems to the classification of their Lie symmetries. We also construct generating sets of such conservation laws under action of Lie symmetries.


Introduction
The shallow water equations play a distinguished role amongst models in geophysical fluid mechanics. They are used both as a starting model for constructing novel numerical schemes that should eventually be implemented for the full set of three-dimensional hydro-thermodynamical equations governing the evolution of the atmosphere-ocean system, as well as a practical model for simulating the evolution of water waves on the open ocean as needed for tsunami prewarnings. See [1,12,17,18,25,26,29,47,53,54,55,60] and references therein for several results for both types of uses of the shallow water equations.
We remind in more details some references on conservation laws of shallow water equations since this is the subject of the present paper. Conservation laws of the system of one-dimensional shallow water equations with flat bottom topography were considered in [21] jointly with a Hamiltonian formulation for this system found in [37,Section II.6.5]; see also [19,Section 2.2]. The case of linearly sloping bottom profile was discussed in [6,7]. Zeroth-order conservation laws for systems of one-dimensional shallow water equations with variable bottom topography in Eulerian coordinates were computed in [3] depending on bottom topography profiles. First-order conservation laws for the one-dimensional variable bottom topography in Lagrangian variables were constructed in [5] and then considered within the framework of Noether's theorem. The conservation laws of the two-dimensional case have previously been considered in [28,32,38] as well as in [46,48], and have been used to construct conservative numerical schemes in [47,50,60]. All of these considerations, however, focus on the case of flat bottom topography. In the present paper, we classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography, which are of the form Here (u, v) is the horizontal fluid velocity averaged over the height of the fluid column, h is the thickness of a fluid column, b = b(x, y) is the bottom topography measured downward with respect to a fixed reference level, which plays the role of a parameter function, and the gravitational acceleration is set to be equal 1 in dimensionless units. See Figure 1 for a graphical representation of the above quantities.
Cases of extensions of the spaces of zeroth-order conservation laws for systems from the class (1) are classified up to equivalence generated by the equivalence group G ∼ of this class. The consideration of this problem was started in [2] but the results obtained there were not exhaustive. Earlier, a preliminary description of the above spaces was presented in [4]. We use an optimized version of the method of furcate splitting, essentially following the proof of group classification of the class (1) in [10]. This is the first application of the method of furcate splitting to classifying conservation laws. Initially, the method was suggested in [39] in the course of the group classification of the class of nonlinear Schrödinger equation and was then applied to group classification of various classes of differential equations [16,33,42,56,57,58,59]. Among listed G ∼ -inequivalent cases of extensions of spaces of zeroth-order conservation laws, there are additional equivalences induced by admissible point transformations within the class (1) that are not generated jointly by elements of G ∼ and by point symmetry groups of systems from the class (1). Families of such admissible point transformations were found in [10]. We extend the well-known Hamiltonian representation for the shallow water equations with flat bottom topography and the known family of distinguished (Casimir) functionals of the corresponding Hamiltonian operator [45,Eqs. (3.6)-(3.7)], see also [48,Section 4.4], to an arbitrary bottom topography, which allows us to relate the classifications of Lie symmetries and of zeroth-order conservation laws for systems from the class (1). To find generating sets of zerothorder conservation laws under the action of Lie symmetries, we study this action and derive a convenient general formula describing the action of generalized symmetries on cosymmetries.
The further organization of the paper is the following. For convenience of references, in Section 2 we present results from [10] on the generalized equivalence group and the equivalence groupoid of the class (1) and on the classification of Lie symmetries of systems from this class. Section 3 contains the preliminary analysis of determining equations for characteristics of zerothorder conservation laws of systems from the class (1) and the statement of classification results on such conservation laws. The proof of the classification is presented in Section 4. Hamiltonian structures of systems from the class (1) are used in Section 5 for interpreting and relating the classifications of Lie symmetries and of zeroth-order conservation laws. Therein we also compute generating sets (see [34] for a definition) of such conservation laws under the action of Lie symmetries for all classification cases that are inequivalent up to point transformations. In the final Section 6 we summarize the findings of the paper and discuss possible future research directions.
The group G ∼ is generated by continuous equivalence transformations for which δ 1 > 0 and ε = 1, and two discrete equivalence transformations alternating signs of variables and being involutions, Then we classified Lie symmetries of systems from the class (1) up to G ∼ -equivalence. For any fixed value of the arbitrary element b, the maximal Lie invariance algebra g b of the system L b consists of the vector fields of the form the parameters F 1 , F 2 and F 3 are smooth functions of t and the parameters c 1 and c 2 are constants that satisfy the classifying equation with F 4 being one more smooth parameter function of t. Up to antisymmetry, the nonzero commutation relations between the above vector fields are exhausted by the following ones: It is convenient to denote D x := 2D(t) − 2D t = x∂ x + y∂ y + u∂ u + v∂ v + 2h∂ h and to use hereafter, simultaneously with (x, y), the polar coordinates (r, ϕ) on the (x, y)-plane, r := x 2 + y 2 , ϕ := arctan y x .
Theorem 2. The kernel Lie invariance algebra of systems from the class (1) is g ∩ = D(1) . A complete list of G ∼ -inequivalent Lie symmetry extensions within the class (1) is exhausted by the following cases, where f denotes an arbitrary smooth function of a single argument, α, β, µ and ν are arbitrary constants with α 0 mod G ∼ , β > 0 and additional constraints indicated in the corresponding cases, ε = ±1 mod G ∼ and δ ∈ {0, 1} mod G ∼ .
Remark 3. The presence of additional equivalences among classification cases listed in Theorem 2 justifies the usage of two-level numeration for them: numbers with the same Arabic numerals and different Roman letters correspond to cases that are equivalent with respect to additional equivalence transformations.

Preliminary analysis and classification result
Suppose that for a fixed value of the arbitrary element b, a tuple γ = (γ 1 , γ 2 , γ 3 ) T of sufficiently smooth functions of (t, x, y, u, v, h) is a characteristic of a conservation law where D † L b is the adjoint of the Fréchet derivative of the left-hand side of the system (1), with D t , D x and D y denoting the total derivative operators with respect to t, x and y, respectively. Since L b is a system of evolution equations, it is natural to assume the derivatives (u t , v t , h t ) as the leading ones and express them in terms of other (parametric) derivatives in view of the system L b , To get the system of determining equations for the components of the tuple γ, we first confine the expanded expression for D † L b γ in the condition (4) to the manifold defined by L b in the jet space J 1 (t, x, y|u, v, h) via substituting the above expressions for the leading derivatives. The derived equations are then split with respect to the first-order (parametric) derivatives of the dependent variables (u, v, h) with respect to x and y. We additionally rearrange the equations obtained after the splitting and exclude those of them that are differential consequences of the others, which finally gives the system The equations γ 1 v = γ 2 u , γ 1 h = γ 3 u and γ 2 h = γ 3 v imply that the Fréchet derivative of γ is a formally self-adjoint operator, which is a necessary and sufficient condition for a cosymmetry of an evolution system to be a conservation-law characteristic of this system. In other words, the space of nonpositive-order cosymmetries of L b coincides with Ch 0 b . The general solution of the subsystem (5a) can be represented in the form where the coefficients F 1 , γ 10 , γ 20 and γ 30 depend at most on (t, x, y). We substitute this representation into the equations (5b)-(5d) and split the resulting equations with respect to (u, v, h) to derive a system for representation's coefficients, Differential consequences of (6a)-(6b) are the equations γ 10 tx = γ 20 ty = 0 and γ i0 xx = γ i0 xy = γ i0 yy = 0, i = 1, 2. Thus, the subsystem (6a)-(6b) integrates to where F i , i = 1, 2, 3, 4, are sufficiently smooth functions of t, and c 1 is a constant. This results in a refined form of the components of the tuple γ, where, as above, F i , i = 1, 2, 3, 4, are sufficiently smooth functions of t, and c 1 is a constant. In other words, for any b where the parameters F 1 , F 2 , F 3 and F 4 run through the set of smooth functions of t, For elements of Ch 0 b , the parameters F 1 , F 2 , F 3 , F 4 and c 1 additionally satisfy the equation implied by (6c), The equation (9) is the only classifying equation for nonpositive-order conservation-law characteristics of systems from the class (1). Thus, the problem of classification of such characteristics up to G ∼ -equivalence reduces to solving the equation (9) up to the same equivalence with respect to the arbitrary element b and the parameters F 1 , . . . , F 4 and c 1 .
can be computed directly from the characteristic form of the condition for conserved currents, "Integrating by parts" in the right-hand side of the last equality and taking into account the classifying equation (9), we obtain that up to adding a trivial conserved current with Recall that r := x 2 + y 2 and ϕ := arctan(y/x) are the polar coordinates (r, ϕ) on the (x, y)-plane, Theorem 4. For any value of the arbitrary element b, the system L b admits two linearly independent conservation laws with characteristics . A complete list of G ∼ -inequivalent extensions of the spaces of nonpositive-order conservation-law characteristics, Ch 0 b , within the class (1) is exhausted by the following cases, where f denotes an arbitrary smooth function of a single argument, α, β, µ and ν are arbitrary constants with α 0 mod G ∼ , β > 0 and additional constraints indicated in the corresponding cases, ε = ±1 mod G ∼ and δ ∈ {0, 1} mod G ∼ .  Similarly to the classification of Lie symmetries, the families T 1 -T 3 of admissible transformations of the class (1) induces additional equivalences among classification cases of Theorem 4, T 3 : 14b → 14a, 4, 8a, 10, 12: 2 δ = 1 → δ = 0. This is why in Theorem 4 we use, analogously to Theorem 2, the two-level numeration of classification cases, cf. Remark 3.
In view of the above equivalences, the following assertions are obvious.
Corollary 7. A system from the class (1) admits a nine-dimensional space of zeroth-order conservation laws if and only if it is reduced by a point transformation to the system from the same class with b = 0.
Corollary 8. Any system from the class (1) with five-dimensional space of zeroth-order conservation laws is similar, with respect to a point transformations, to the system from the same class with b = ±r −2 .
Remark 9. The comparison of Theorems 2 and 4 shows that the extension of the space of zeroth-order conservation laws is possible only for systems with Lie-symmetry extension. The correspondence of cases of Theorem 4 to those of Theorem 2 is obvious, and there is a correlation between the dimensions of the corresponding spaces. Cases 1 ν =−2 , 2, 4, 5, 10, 11, 12a-12c with α = 0, 13, 14 and 15 of Theorem 2 have no counterparts among cases of Theorem 2. All these facts can be explained within the framework of the Noether relation between Hamiltonian symmetries and conservation laws for systems from the class (1), see Section 5.
Using the above correspondence, we can adopt Conjecture 13 from [10], which concerns the group classification of the class (1) up to the general point equivalence and was justified by proving the major part of (in)equivalences among classification cases by algebraic tools, to zeroth-order conservation laws.
In view of the reduction of the number of classification cases for zeroth-order conservation laws in comparison with that for Lie symmetries, for proving Conjecture 10 we just need to verify the inequivalence of Cases 1 and 2 and the impossibility of further gauging of constant parameters remaining in some cases. This requires the complete description of the equivalence groupoid of the class (1), which is still unknown.
Remark 11. Among constructed conservation laws, there are those with obvious physical interpretation. Thus, the characteristics Λ 0 , Λ 1 b ( 1 2 ), Λ 2 (1), Λ 3 (1) and Λ 4 (1) corresponds to the conservation of angular momentum, energy, x-momentum, y-momentum and mass, respectively. Note the zeroth-order conservation laws associated with distinguished (Casimir) functionals of the Hamiltonian operator for systems from the class (1) are exhausted by those with characteristics of the form Λ 4 (c), where c is an arbitrary constant, see Section 5 below.

Proof of the classification
For classifying nonpositive-order conservation-law characteristics of systems from the class (1), the method of furcate splitting works in the same way as for classifying Lie symmetries of such systems, i.e., via fixing an arbitrary value for the variable t in the classifying equation (9), which results in the so-called template form of equations with respect to the arbitrary element b: Here, a 1 , . . . , a 8 are constants. The maximal number k = k(b) of template-form equations with linearly independent coefficient tuplesā i = (a i 1 , . . . , a i 8 ), i = 1, . . . , k, that hold for a fixed value of the arbitrary element b, depends on the fixed value but hereafter we will not indicate this dependence explicitly. It is clear that 0 k 8. If k > 0, then we denote the matrix with coefficients of the system of template-form equations (12) and its submatrix constituted by the first l columns of A by Then we have v 1 , . . . , v k ∈ a, where Direct computation shows that . . , k. Therefore, the span a is closed with respect to the Lie bracket of vector fields, thus constituting a Lie algebra. The equation for b associated with [v i , v i ] is a differential consequence of the subsystem of the equations associated with v i and with v i , and it is of the same template form (11). Since the maximal number of linearly independent vector fields corresponding to template-form equations for the respective value of the arbitrary element b coincides with k = k(b), we obtain It is also possible to use the condition (14) formulated for the projectionsv 1 , . . . ,v k of the vector fields v 1 , . . . , v k to the space with the coordinates (x, y), which reads The system (12) can be simplified by linearly combining its equations and acting by transformations from G ∼ . Specifically, in the case a i j = 0 one can divide the ith equation of (12) by a i j to set a i j = 1. Using simultaneous shifts with respect to x and y, which belong to G ∼ , one can set either a i 3 = a i 4 = 0 if (a i 1 , a i 2 ) = (0, 0) or a i 6 = a i 7 = 0 if a i 5 = 0. Similarly, if a i 1 = 0, then shifts of b allows one to set a i 8 = 0. The value k = 0 corresponds to the case of unrestricted arbitrary element b. Here, one can split the equation (9) with respect to b and its derivatives, which yields F 1 t = F 2 = F 3 = F 4 t = 0 and c 1 = 0. Thus, for any value of the arbitrary element b, the system L b admits two linearly independent conservation laws with characteristics Λ 1 b (1) and Λ 4 (1), i.e., . For each of the other distinguished cases k = 1, . . . , k = 4, below we make the following steps, additionally splitting the consideration into subcases depending on values of the parameters a i j : • for k > 1, compute the values of the parameters a i j for which the related system (12) is compatible and can be derived by splitting the equation (9), • gauge maximally possible number of the parameters a i j by linearly recombining templateform equations and by acting transformations from the group G ∼ and re-denote the remaining parameters a i j , • integrate the reduced system (12) of template-form equations with respect to the arbitrary element b, • gauge, if possible, the integration constant by equivalence transformations of the class (1), • solve the system of determining equations with respect to the parameters c 1 , F 1 , F 2 and F 3 , which results in the spaces of nonpositive-order conservation-law characteristics, Ch 0 b , of systems from the class (1) with obtained values of b.
The order of steps is not fixed, and they can intertwine. In view of (10), the construction of the corresponding spaces of canonical conserved currents is straightforward.

One independent template-form equation
If k = 1, then the right-hand side of the classifying equation (9) is proportional to the righthand side of the single template-form equation (12), where the proportionality coefficient λ is a sufficiently smooth function of t, i.e.

Two independent template-form equations
If k = 2, then the right-hand side of the classifying equation (9) is represented as a linear combination of right-hand sides of the first and the second equations of the system (12) with coefficients λ 1 and λ 2 that are functions of t and depend on the parameters involved in elements of Ch 0 b , Remark 12. The coefficients λ 1 and λ 2 cannot be proportional to each other with the same constant multiplier for all elements of Ch 0 b . Indeed, otherwise, i.e., if µ 1 λ 1 + µ 2 λ 2 = 0 for some constants µ 1 and µ 2 with (µ 1 , µ 2 ) = (0, 0), there would be no additional conservation-law extensions as compared to the more general case of conservation-law extensions with k = 1, and the linear combination of the equations (12) with coefficients µ 2 and −µ 1 would play the role of a single template-form equation. For the same reason, the coefficients λ 1 and λ 2 are simultaneously nonvanishing for some elements of Ch 0 b .
Similar to the case k = 1, we can split the above condition with respect to the arbitrary element b, its derivatives b x and b y and the independent variables x and y, obtaining the system Depending on the rank of the submatrix A 2 , which consists of the first two columns of A and thus rank A 2 2, the further study of the case k = 2 splits into three subcases, rank A 2 = 2, rank A 2 = 1 and rank A 2 = 0 rank A 2 = 2. We linearly recombine equations of the system (12) to set A 2 to the 2 × 2 identity matrix or, equivalently, a 1 1 = a 2 2 = 1 and a 1 2 = a 2 1 = 0. For further simplification of the system (12), we set a 1 3 = a 1 4 = a 1 8 = 0 by shifting x, y and b, respectively. The vector fields v 1 and v 2 , which are associated with the first and the second equations of the reduced system (12), respectively, commute in view of the condition (14), which leads to the following system of algebraic equations for the coefficients a i j : The system (17) then reduces to F 1 t = λ 1 , c 1 = λ 2 , F 2 = 0, F 3 = 0, λ 1 tt + a 1 5 λ 1 = 0, a 1 6 λ 1 + a 2 6 λ 2 = 0, a 1 7 λ 1 + a 2 7 λ 2 = 0, F 4 t = 0.
Remark 12 implies that the sixth and the seventh equations of the system (19) are equivalent to a 1 6 = a 2 6 = 0 and a 1 7 = a 2 7 = 0, respectively. The system (12) reduces in the polar coordinates (r, ϕ) to rb r + 2b + 1 2 a 1 5 r 2 = 0, b ϕ = 0, with the parameter a 1 5 belonging to {0, −4, 4} (up to scaling equivalence transformations). The general solution of the last system is where we should assume the integration constant b 0 to be nonvanishing since the above value of b with b 0 = 0 is related to the case k = 4. We can therefore scale b 0 by an equivalence transformation to ε = ±1 and obtain, depending on the value of the parameter a 1 5 , Cases 7a, 7b and 7c of Theorem 4. rank A 2 = 1. We begin by linearly recombining equations of the system (12), thereby reducing the matrix A 2 to In other words, a 2 1 = a 2 2 = 0 and (a 1 1 , a 1 2 ) = (0, 0). Since rank A 4 = 2, we also get (a 2 3 , a 2 4 ) = (0, 0). From the condition (14) it follows that a 1 2 = 0 and hence a 1 1 = 0. We can thus set a 1 1 = 1 by re-scaling the first equation, a 2 3 = 1, a 2 4 = 0 by a rotation equivalence transformation and rescaling the second equation as well as a 1 3 = a 1 4 = a 1 8 = 0 by shifts of x, y and b. The commutation relation [v 1 , v 2 ] = −v 2 , which is derived from the condition (14), yields the following system of algebraic equations for the remaining coefficients a i j : The system (17) reduces to After recalling Remark 12, we get from the eighth equation of the system (21) that a 1 7 = 0. For a 1 5 = 0, shifting x and b and recombining equations (12), we can set a 1 6 = 0. Then a 2 8 = 0. The integration of the system (12) gives the expression b = b 0 y −2 − 1 8 a 1 5 r 2 , where again the integration constant b 0 does not vanish in view of the assumption k = 2 as otherwise this value of the arbitrary element b is associated with k = 4. Scaling equivalence transformations allow us to set b 0 , a 1 5 /4 ∈ {−1, 1}. As a result, we derive Cases 8b and 8c of Theorem 4 for a 1 5 = −4 and a 1 5 = 4, respectively. For a 1 5 = 0, we also have a 2 6 = 0. Integrating the system (12), we obtain b = b 0 y −2 − a 2 8 x, where again the integration constant b 0 is nonzero in view of the same argument as above. Up to scaling equivalence transformations and alternating the signs of (x, u), we can make b 0 ∈ {−1, 1} and a 2 8 ∈ {0, −1}, which leads to Case 8a. rank A 2 = 0. In other words, a 1 1 = a 1 2 = a 2 1 = a 2 2 = 0. Because rank A 4 = 2, equations of the system (12) can be linearly recombined for getting a 1 3 = a 2 4 = 1 and a 1 4 = a 2 3 = 0. The compatibility condition (14) implies that the vector fields v 1 and v 2 associated to equations of the reduced system (12) commute, i.e. [v 1 , v 2 ] = 0. This condition is equivalent to the system a 1 5 = a 2 5 = 0, a 1 7 = a 2 6 .
Due to the last equation, we can set a 1 7 = a 2 6 = 0 by rotation equivalence transformations. Then the system (12) takes the form b x + a 1 6 x + a 1 8 = 0, b y + a 2 7 y + a 2 8 = 0. Its general solution is where b 0 is an integration constant. Therefore, a 1 6 = a 2 7 since otherwise this form of the arbitrary element b is related to the value k = 4, which contradicts the supposition k = 2. We can set b 0 = 0 modulo equivalence transformations of shifts with respect to b. Up to G ∼ -equivalence, we can also set a 2 6 = ±1, a 2

More independent template-form equations
We show below that in fact the case k = 3 is not possible and thus k = 4 if k > 2. k = 3. The condition rank A 4 = rank A = k = 3 necessarily implies that rank A 2 > 0.
The assumption rank A 2 = 2 results in a contradiction. Indeed, under this assumption we can linearly recombining equations of the system (12) to reduce the submatrix A 2 to the form After the reduction, the projections of the vector fields v 1 , v 2 and v 3 to the space with the coordinates (x, y) are given bŷ The condition (15) requires that the commutator [v 2 ,v 3 ] = −a 3 4 ∂ x + a 3 3 ∂ y belongs to the span v 1 ,v 2 ,v 3 but this is not the case.
Hence rank A 2 = 1, and consequently the linear recombination of equations (12) reduces the matrix A 4 to the form Then the compatibility condition (14) is expanded to the commutation relations The last commutation relation implies a 2 5 = a 3 5 = 0 and a 3 6 = a 2 7 . Modulo rotation equivalence transformations, we can set a 2 7 = a 3 6 = 0. The former commutation relations together with the last constraint imply the equations a 1 1 (a 2 6 −a 3 7 ) = 0 and a 1 2 (a 2 6 −a 3 7 ) = 0. In view of (a 1 1 , a 1 2 ) = (0, 0), we then derive a 3 7 = a 2 6 . In view of the equations b x + a 2 6 x + a 2 8 = 0 and b y + a 2 6 y + a 3 8 = 0, the arbitrary element b is necessarily a quadratic polynomial of (x, y) with the same coefficients of x 2 and of y 2 and with zero coefficient of xy. It is obvious that each such quadratic polynomial satisfies a system of the form (12) with k = 4, which contradicts the assumption k = 3. k = 4. Taking into account the condition rank A 4 = rank A = 4, we linearly recombine the equations (12) to reduce A 4 to the 4 × 4 identity matrix. Due to simultaneously reducing the form of the vector fields v 1 , . . . , v 4 , the compatibility condition (14) in particular gives the commutation relations The last commutation relation leads to the equations a 3 5 = a 4 5 = 0 and a 3 7 = a 4 6 . Revisiting the first two commutation relations, we derive a 2 5 = −2a 3 7 = 2a 3 7 , a 3 6 = a 4 7 , and thus a 3 7 = 0. As a result, the subsystem of the last two equations (12) takes the form b x + a 3 6 x + a 3 8 = 0, b y + a 3 6 y + a 4 8 = 0. Any solution of this subsystem is a quadratic polynomial of (x, y) with the same coefficients of x 2 and of y 2 and with zero coefficient of xy. Moreover, k = 4 for all such values of b, and only four of them are G ∼ -inequivalent, b = 0, b = x, b = 1 2 (x 2 + y 2 ) and b = − 1 2 (x 2 + y 2 ). which correspond to Cases 14a, 14b, 14c and 14d of Theorem 4, respectively.

Hamiltonian structures and generating sets of conservation laws
Hamiltonian representations for systems from the class (1) can be easily constructed by generalizing the well-known Hamiltonian representation for the shallow water equations with flat bottom topography [45, Eqs. (3.6)-(3.7)], see also [48,Section 4.4]. For each value of the arbitrary element b, the system L b is Hamiltonian and can be represented in the form w t = H δH b , where δ stands for the variational derivative with respect to the tuple of dependent variables w = (u, v, h), are the associated Hamiltonian differential operator and the associated Hamiltonian functional, D x and D y denote the total derivative operators with respect to x and y, respectively, and q := (v x − u y )/h is the shallow water potential vorticity. Thus, The local conservation law associated with the functional C R has the characteristic T (cf. the last equation) and contains the conserved current hR(q) (1, u, v). Therefore, this conservation law is zero if and only if R = cq for some constant c. Its order is equal to zero or one if R = 0 or R = 0, respectively. Among elements of the family {C R }, there are functionals associated with the conservation of mass (R = 1), the trivial conservation of circulation (R = q) and the conservation of potential enstrophy (R = 1 2 q 2 ). The operator H maps cosymmetries and conservation-law characteristics of L b to characteristics of symmetries and of Hamiltonian symmetries, respectively. Acting by H on the elements (8) of Ch 0 , we obtain The tuple Λ 4 (F 4 ) belongs to the kernel of H since it is the variational derivative of the timedependent distinguished functional F 4 C 1 . At the same time, the tuple coinciding with D t [w] on solutions of L b does not belong to the image of H under acting on triples of differential functions since, in particular, h cannot be equal to the total divergence of a pair of differential functions of w with respect to the space variables (x, y).
This means that the algebra h b = HCh 0 b of Hamiltonian Lie symmetries of the system L b coincides with the intersection g b ∩ h , where h := D(F 1 ), J, P (F 2 , F 3 ) with the parameter functions F 1 , F 2 and F 3 running through the set of smooth functions of t. Note that h b is an ideal of g b with dim h b = dim Ch 0 b − 1, and HCh 0,unf b = g ∩ , HCh 0 ∩ = {0}. Using H as a Noether operator, we can endow the space Ch b of cosymmetries of the system L b with structure of Lie algebra, where Lie bracket [·, ·] H is defined by for the canonical representatives γ 1 and γ 2 of arbitrary cosymmetry cosets of L b , cf., e.g., [30] and [15,Section 3.1]. (The canonical representative of a cosymmetry coset of a system of evolution equations is the element of the coset that depend on no derivatives involving differentiation with respect to t.) Recall that D γ and D † γ denote the Fréchet derivative of γ and its adjoint, respectively. Formally commuting the elements (8) of Ch 0 , we get Therefore, the space Ch 0 b of characteristics of zeroth-order conservation laws is closed with respect to the above bracket in view of the classifying equation (9). Moreover, for any γ 1 , γ 2 ∈ Ch 0 b we have [γ 1 , γ 2 ] H = D γ 2 Hγ 1 + D † Hγ 1 γ 2 since D γ = D † γ for the canonical representative γ of any coset of conservation-law characteristics of L b . In other words, the operator H gives rise to a homomorphism from the Lie algebra Ch 0 b to the Lie algebra g b with the kernel Λ 4 (1) and the The above consideration establishes the Noether relation between Hamiltonian Lie symmetries and zeroth-order conservation laws for each system from the class (1). Moreover, this consideration explains the facts given in Remark 9. An extension of the space Ch 0 b corresponds to an extension of the algebra h b of Hamiltonian Lie symmetries rather than an extension of the entire algebra g b of Lie symmetries. For values of the arbitrary element b, where h b is a proper subalgebra of g b , the correspondence between g b and Ch 0 b is broken. Since in Cases 1 ν =−2 , 2, 4, 5, 10, 11, 12a-12c with α = 0, 13, 14 and 15 of Theorem 2, vector fields extending the corresponding algebra g b involve D t and are thus not Hamiltonian, these cases have no counterparts among cases of Theorem 4. In Cases 17-21 of Theorem 2, the algebra h b extends although it is a proper subalgebra of g b . Hence these cases have counterparts among cases of Theorem 4 but dim g b > dim Ch 0 b − 1. One could single out Hamiltonian symmetries among Lie symmetries listed in Theorem 2 and then use them for constructing zeroth-order conservation laws of systems from the class (1) within the framework of the Noether relation between Hamiltonian symmetries and conservation laws [40,Theorem 7.15]. This would allow to classify zeroth-order conservation laws for systems from the class (1) using the group classification of this class in Theorem 2. At the same time, the direct classification of such conservation laws in Section 4 seems to be a more efficient way. Now for each value of the arbitrary element b, we study the action of Lie symmetries of the system L b on its zeroth-order conservation laws, more precisely, on characteristics of such conservation laws. 3 Thus, we can construct a generating set for the corresponding space, which leads to a generating set for zeroth-order conservation laws. 3 Let L be a system of p differential equations L s (x, u (r) ) = 0, s = 1, . . . , p, for m unknown functions u = (u 1 , . . . , u m ) of n independent variables x = (x1, . . . , xn). Here u (r) denotes the set of all the derivatives of the functions u with respect to x of order not greater than r, including u as the derivative of order zero. Suppose that a generalized vector field Q = n i=1 ξ i [u]∂x i + m a=1 η a [u]∂ua is a generalized symmetry of L, and a tuple γ = (γ 1 [u], . . . , γ p [u]) T is a cosymmetry (resp., more specifically, a conservation-law characteristic) of L. Here ξ i [u], η a [u] and γ s [u] are differential functions of u. Therefore, Q (r) L s = p s =1 Υ ss L s for some matrix operator Υ = (Υ ss ) s,s =1,...,p whose entries are differential operators in total derivatives with respect to x with coefficients being differential functions of u. Here Q (r) is the rth prolongation of Q. See [40] for definitions. The action of Q on γ is given by LQγ = (Q (∞) + Div ξ)γ + Υ † γ, where Div ξ := n i=1 Diξ i is the total divergence of ξ := (ξ 1 [u], . . . , ξ n [u]), Di is the total derivative operator with respect to xi, i = 1, . . . , n, and the matrix operator Υ † is the formally adjoint of Υ.
Hence the action of Q on a zeroth-order cosymmetry γ of L b is given by Acting by L Q on a tuple γ =c 1 Λ 0 +Λ 1 b (F 1 )+Λ 2 (F 2 )+Λ 3 (F 3 )+Λ 4 (F 4 ) ∈ Ch 0 b with functionsF 1 , F 2 ,F 3 andF 4 of t and a constantc 1 , we get tF a deeper understanding of them. Due to the possibility of following the proof of group classification, the direct construction of zeroth-order conservation laws of systems from the class (1) is of the same order of computational complexity as the construction based on the Noether relation between Hamiltonian symmetries and conservation laws. An incidental benefit of the direct construction is a test of furcate splitting as a method for classifying low-order conservation laws that can be applied to classes of non-Hamiltonian systems as well.
A Lie structure underlying the template-form equations for conservation laws of each system from the class (1) provided us with algebraic tools for optimizing the procedure of furcate splitting. It is not clear yet whether this Lie structure is specific for systems from the class (1) and induced by their Hamiltonian structure or whether similar Lie structure may arise in the course of classification of conservation laws for non-Hamiltonian systems.
Along with zeroth-order conservation laws, each system from the class (1) admits an infinite number of linearly independent first-order conservation laws associated with distinguished (Casimir) functionals of the Hamiltonian operator H, which is common for all such systems. An interesting question is whether the above conservation laws span the entire space of local conservation laws for any system from the class (1).