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Completeness is determined by any non-algebraic trajectory

It is proved that any polynomial vector field in two complex variables which is complete on a non-algebraic trajectory is complete.


Introduction and Statement of Results
Let X be a holomorphic vector field on C 2 . For any z ∈ C 2 , the local solution ϕ z (T ) of the associated ordinary differential equation dz/dT = X(z(T )) with the initial condition z(0) = z ∈ C 2 can be extended by analytic continuation along paths in C, to a maximal domain Ω z , which may not be an open set of C, but rather a Riemann domain over C. The map ϕ z : Ω z → C 2 is the solution of X through z, and its image ϕ z (Ω z ), that will be denoted by C z (or L z , R z , S z ), is the trajectory of X through z.
The vector field X is complete on C z if Ω z = C, and X is complete if it is complete on C z , for every z ∈ C 2 . Each trajectory C z on which X is complete (complete trajectory) is defined by an abstract Riemann surface uniformized by C, and by the maximum principle, analytically isomorphic to C or C * .
Extrinsically, the topology of a trajectory can be very complicated. The simplest trajectories from this point of view are the analytic ones. One says that the trajectory C z is analytic if it is contained in an analytic curve in C 2 (but not necessarily equal to it, due to the possible presence of singularities). Otherwise C z is a non-analytic trajectory.
An interesting remark (due to R. Moussu) is that two vector fields with a common non-analytic trajectory have to be collinear in any point. In this sense a non-analytic trajectory determines the vector field up to multiplication by a nonvanishing holomorphic function.
In this work we will consider polynomial vector fields with at most isolated zeros. The above remark for two polynomial vector fields can be restated.
For a trajectory, it is enough not to be contained in an algebraic curve (that is, to be a non-algebraic trajectory) to determine the vector field up to multiplication by a nonzero constant.
In [1], M. Brunella studied foliations in C 2 given by polynomial vector fields with a trajectory containing a planar isolated end (proper Riemann sub-surface isomorphic to {z : r < ||z|| ≤ 1}, where r ∈ [0, 1)), properly embedded in C 2 and whose closure in CP 2 contains the line at infinity. He proved that these foliations can be determined in terms of a polynomial whose generic fiber is of type C or C * and transversal to the foliation. As remarkable corollary, he obtained that if the trajectory is a non-algebraic analytic plane, the foliation is given by the constant vector field after an analytic automorphism. So the trajectory in this case is determining the completeness of the vector field up to multiplication by a nowhere vanishing function.
Then if one attends to the completeness of a non-algebraic trajectory (not necessarily analytic) the following natural question arises [6, Question 5.1]: Question 1. If X is a polynomial vector field in C 2 with the property of being complete on a single non-algebraic trajectory, is it complete?
The main result of this work says that Question 1 has an affirmative answer: Theorem 1. Let us consider a polynomial vector field X on C 2 which is complete on a non-algebraic trajectory. Then X is complete.
Note that our theorem implies that any entire solution of a polynomial vector field can be determined up to an algebraic automorphism of C 2 . As the vector field is complete, the solution must correspond to one of the vector fields of the Brunella's classification in [4] after a polynomial automorphism.
It could be very interesting to study if a non-analytic trajectory of a (non-polynomial) holomorphic vector field determines the completeness of the vector field.
About the proof of Theorem 1. For the sake of completeness, throughout the paper we include some definitions and results taken from Brunella's papers [2], [3] and [4]. Let us begin by recalling some definitions. Consider the foliation F generated by X on C 2 extended to CP 2 . According to Seidenberg's Theorem, the minimal resolution of F is a new foliationF defined on a rational surface M after pulling back F by a birational morphism π : M → CP 2 , that is a finite composition of blowing ups. Along with this resolution one has: 1) The Zariski open set U = π −1 (C 2 ) of M , over which X can be lifted to a holomorphic vector fieldX, 2) the exceptional divisor E of U , and 3) the divisor at infinity that is a tree of smooth rational curves. The vector fieldX can be extended to M , although it may have poles along one or more components of D. Let us still denote this extension byX. As only singularities of the foliation in C 2 are blown up, and they are in the zero set of X, the vector fieldX is holomorphic on the full U and it has the complete trajectoryC z defined by π −1 (C z ).
Therefore the reduced foliationF has at least one tangent entire curve: the one defined byC z , which is Zariski dense in M . It implies that the Kodaira dimension kod(F ) ofF is 1 or 0 [13,§IV] (see also [2, p. 131]).
In case kod(F) = 1, [4] allows to conclude thatF is a Riccati foliation adapted to a fibration g : M → P 1 , whose projection to C 2 by π defines a rational function R of type C or C * . We can apply the study of [6] although R is not a polynomial (see also [8]) and deduce the completeness of X. We will analyze this case in §2.
In case kod(F) = 0, we know thatF is generated by a vector field on a smooth compact projective surface S, up to contractions ofF-invariant curves and covering maps [4]. However, we need to go a bit further to know if these models restrict to our open U a complete vector field. This is accomplished via the description of the irreducible components of D ∪ E that are notF-invariant. When S is rational, we show that in fact D ∪ E must be invariant ifF is not Riccati with respect to a fibration g : M → P 1 that is projected to C 2 by π as a rational function R of type C or C * . For the remaining cases, i.e. when S is a CP 1 -bundle over an elliptic curve or a complex 2-torus, we prove that D∪E is always invariant byF . For the proof of this last fact we will consider S as a differential manifold with a certain Riemannian metric. It will enable us to compute the distance from the complete trajectory to a compact set containing the components of D ∪ E that are notF-invariant. As a consequence of the discussion above one obtains that the lifted ofX by a certain covering map can be decomposed in the product of a complete rational vector field by a second integral of it. It allows us to conclude that the projection π * X restricted to U i.e. X must be complete. We will analyze this case in §3.
Finally, we point out that [6], [7] imply that Question 1 has an affirmative answer for a non-algebraic analytic trajectory. In those works, Brunella's results [1] are used as the main tool. The proof of our theorem is mainly based on Brunella's approach to the classification complete polynomial vector fields in the plane [4], since they can be applied to the foliation F although X could be in principle not complete. Theorem 1 is not only the generalization of the previous results mentioned above ([6], [7]), but its proof also implies them.
Acknowledgements. This article is dedicated to the memory of Marco Brunella, whose deep mathematical work about complete vector fields has been crucial to obtain our results. We also appreciate a lot his generosity in several mathematical conversations during these last years. We received the sad notice of his death when we were preparing the revised version of this article.
Finally, we also want to thank the referee for his suggestions that have improved this paper a lot.

kod(F)=1
According to [13,§IV] the absence of a first integral implies thatF is a Ricatti or a Turbulent foliation, that is to say, the existence of a fibration g : M → B whose generic fibre is a rational curve or an elliptic curve transverse toF, respectively. Remark that B is CP 1 since M is a rational surface. Existence of a nef model. AsF is not a rational fibration it has a modelF which is reduced and nef. More concretely, after a contraction s : M →M of theF-invariant rational curves on M over which the canonical bundle KF has negative degree one obtains (see [3, § 4] 1) A new surfaceM , maybe with cyclic quotient singularities; and 2) A reduced foliationF = s * F onM such that its canonical Q -bundle KF is nef (i.e. KF · C ≥ 0 for any curve C ⊂M ).
Recall that a cyclic quotient singularity p ofM is locally defined by B 2 /Γ k,h where B 2 ⊂ C 2 is the unit ball and Γ k,h is the cyclic group generated by a map of the form (z, w) → (e 2πi k z, e 2πi k h w) with k, h positive coprime integers such that 0 < h < k. These singularities ofM are not singularities ofF. That is, the foliation can be lifted locally to B 2 \ {(0, 0)} and extended to a foliation on B 2 with a non-vanishing associated vector field. Existence of a minimal model. After possibly additional contractions onM of rational curves, q :M → N , one obtains a reduced foliation H = q * F (birational toF ) on a surface N regular on the (cyclic quotient) singularities of N whose canonical bundle K H is nef and such that it verifies this property: if K H · C = 0 ⇒ C 2 ≥ 0 for any curve C ⊂ N . It is important to note that we can assume that q is given by contractions of curves which are invariant by the foliation: if C is notF-invariant it follows from the formula Remark 3. As s and q are given by contractions of rational curves which are invariant by the foliation neitherC z meets the exceptional divisor of s nor s(C z ) meets the exceptional divisor of q. It implies that there must be a parabolic leaf of H: the leaf that contains the Riemman surface q(s(C z )) that supports the complete vector field q * (s * (X |Cz )).

Turbulent case.
When X is complete the case of a TurbulentF can be excluded as it is proved in [4,Lemma1]. We now prove that it continues being valid in a more general situation.

Lemma 1.F is not a Turbulent foliation
Proof. Suppose thatF is Turbulent. The description of models around each fibre of g after a birational morphism α : M → M * is known [3, § 7]. The resulted foliation G = α * F on M * is regular on the (cyclic quotient) singularities of M * , it is Turbulent with respect toḡ = g • α −1 , and each fiber ofḡ is of one of the following classes: (a) (resp. (d)): the fibre is smooth elliptic, transversal (resp. tangent) to G and may be multiple.
For any leaf L of G outside tangent fibres ofḡ,ḡ |L : L → B 0 , with B 0 defined as B minus the points over tangent fibres ofḡ, is a regular covering (in orbifold's sense). The orbifold structure in B 0 is the natural structure inherited from the orbifold structure on B induced by (the local models of) g [3, § 7]. Claim 1: There must be at least one tangent fibre G 0 ofḡ.
We suppose that all the fibres are transversal and obtain a contradiction. Since B 0 = B = CP 1 , the orbifold universal covering of any leaf L,L, is equal to the one of B,B.
Let us suppose thatB is C or CP 1 . By pulling back sections of K B underḡ we obtain sections of K G . We can in this way compute K G and obtain that deg [14]). ThusB is a disk.
As all the leaves of G are hyperbolic and the singularities are isolated (in fact G is regular), K G is nef [5,Remark 8.8]. Moreover, it is clear that § 2] implies that C 2 = 0 because G is regular on C. Therefore G is a minimal model ofF. But then it has necessarily a parabolic leaf (Remarks 2 and 3), which is a contradiction.
It is important to note that the strict transform of D 1 by α, that we also denote by D 1 , is a rational curve. Otherwise it is a point with infinitely many punctured disks invariant by G through it and then a singularity of G, which is not possible. Hence D 1 ∩ G 0 = ∅. Let us denote by J the leaf of G that defines the non algebraic component of α(C z ). There is at least one accumulation point of J on G 0 becauseḡ(J) = g(C z ) is C or C * . It must be a regular point of the foliation by the absence of singularities of the foliation on tangent fibers. Thus J must accumulate on G 0 . It implies that C z ∩ D 1 = ∅ and then D 1 ⊂ {X = 0} by the completeness ofX |Cz .
Let us take the generic fiber G of g, which is transverse toF . Obviously, D ∩ G = ∅. In the contrary case we have an elliptic curve contained in C 2 , which is impossible (maximum principle). Among the irreducible components of D cutting G at least one, say D 2 , isF -invariant. OtherwiseX would be holomorphic in a neighborhood of G and it vanishes on at least one component of D transversal to G, what implies thatX is identically zero by Claim 2. The existence of D 2 is enough to construct a rational integral for a TurbulentF as it can be seen in [4, p.438].

Ricatti case.
Lemma 2. g |U is projected by π as a rational function R of type C or C * . Moreover,F is R -complete.
Proof. Up to contraction of rational curves inside fibers of g, which can produce cyclic quotient singularities of the surface but on which the foliation is always regular, one has that there are five possible models for the fibers of g [3, § 7], [4, p. 439]. Let L 0 be the leaf of the foliation defined byC z . One can conclude that the orbifold universal coveringL 0 of L 0 is equal to the one of B 0 ,B 0 , where B 0 is defined as CP 1 minus the points over tangent fibres of g with the natural orbifold structure inherited from the orbifold structure on CP 1 induced by (the local models of) g. Since X is complete on C z ,L 0 is biholomorphic to C and then L 0 is parabolic. This fact along with kod(F ) = 1 implies by [4, Lemma 2] that there must be at least one fibre G 0 tangent to the foliation of class: (d): the fibre is rational with two saddle-nodes of the same multiplicity m, with strong separatrices inside the fibre, or of class (e): the fibre is rational with two quotient singularities of order 2, and a saddle-node of multiplicity l, with strong separatrix inside the fibre.
Firstly one observes that there are irreducible components of D ∪ E that are not contained in any fiber of g. Let us take the generic fiber G of g, which is transverse toF . Obviously, D∩G = ∅. In the contrary case we have a rational curve contained in C 2 , which is impossible (maximum principle).
Let D 1 be one of these components. Then D 1 ∩ G 0 = ∅ and there is at least one accumulation point ofC z on G 0 , say p, because g(C z ) is C or C * . If p is a regular point of the foliation,C z must accumulate on G 0 . If p is singular, it is a saddle-node with strong separatrix defined by G 0 , and thereforeC z must also accumulate on all G 0 [12], in particular in the other saddle node if it exists. There are two possibilities: (i) If D 1 isF -invariant, D 1 is not in the divisor of poles ofX. Otherwise, D 1 ∩ G 0 = ∅ is a saddle-node q. Let us take the rational section ω of KF dual toX that restricts toC z as the differential of times given by the flow ofX onC z . One can construct a path γ : (0, ǫ] →C z , with ǫ ∈ R + and γ(t) → q as t → 0, such that γ ω is finite (see [4, proof of Lemma 3]), which contradicts the completeness ofX onC z .
It follows from (i) and (ii) thatX is holomorphic in a neighborhood of G, what implies as in above lemma that (ii) does not really occurs. Thus Therefore D must cut G at one or two points, and the projection R of g |U via π is of type C or C * . Moreover, the invariancy of the components of D ∪ E which are not contained in fibers of g implies that generically R is a fibration trivialized by the leaves ofF, and thenF is R-complete.
We will study the two possibilities after the previous lemma.
2.4. R of type C. By Suzuki (see [17]) we may assume that R = x, up to a polynomial automorphism. Hence F is a Riccati foliation adapted to x and X is a complete vector field of the form with C ∈ C, N = 0, 1 and A, B ∈ C[x] (see [6,Proposition 4.2]).
New coordinates. According to relations x = u n and x ℓ y + p(x) = v u −m it is enough to take the rational map H from u = 0 to x = 0 defined by Although R is not necessarily a polynomial (n ∈ Z), it is a consequence of the proof of [6, Proposition 3.2] that H * F is a Riccati foliation adapted to v n having u = 0 as invariant line. Thus where k ∈ Z, and a, c ∈ C [v].
At this point one could apply the techniques of [7] to analyze the possible global 1-forms of times associated to X in order to prove the existence of an invariant line. However, applying directly the local models of [4], it follows from [8, Lemma 2] that at least one of the irreducible components of R over 0 must be a F-invariant line. Hence the polynomial c(v) of (3) is in fact a monomial, and thus of the form cv N with c ∈ C and N ∈ N.

kod(F)=0
According to [13, §III and §IV] we can contractF-invariant rational curves on M via a contraction s to obtain a new surfaceM (maybe singular with cyclic quotient singularities), a reduced foliationF on this surface, and a finite covering map r from a smooth compact projective surface S toM such that: 1) r ramifies only over cyclic (quotient) singularities ofM and 2) the foliation r * (F) is generated by a complete holomorphic vector field Z 0 on S with isolated zeroes [4, p. 443]. Remark 4. Note thatC z does not meet the exceptional divisor of the contraction s. Let us setĈ z as s(C z ). SinceĈ z does not contain singularities ofM thenĈ z is a Riemann Surface, s |Cz :C z →Ĉ z is a biholomorphism and r |r −1 (Ĉz) : r −1 (Ĉ z ) →Ĉ z is a non-ramified finite covering map. Thus s * (X | Cz ) is complete onĈ z and r * (s * (X |Cz )) is complete on the connected Hence each M i is a Riemann Surface contained in a complete trajectory T z of Z 0 that supports the complete vector field r * (s * (X |Cz )) |M i , which does not necessarily coincide with Z 0|M i . It is convenient to observe that if T z is isomorphic to C * then, necessarily M i = T z , and the vector field r * (s * (X |Cz )) |M i coincides with Z 0 on T z , up to a multiplicative constant. The discrepancy between the two complete vector fields can occur only if T z is isomorphic to C, in which case r * (s * (X |Cz )) |M i could have one (and only one) zero at some point p = T z \ M i .
It follows from [4, p. 443] that the covering r can be lifted to M via a birational morphism g : W → S and a ramified covering h : Let Y be the lift of Z 0 on W via g. Then Y must be a rational vector field on W generating the foliationF given by g * (r * (F )) = h * F . On the other hand,F is also generated by the rational vector fieldX on W given by h * X . Hence there is a rational function F on W such that Remark 5. We remark from the above construction: 1) The map g is a composition of blowing-ups over a finite set Θ = {θ i } s i=1 ⊂ S of regular points of Z 0 . The poles of Y are in g −1 (Θ) and they define a divisor P ⊂ W invariant byF . Hence Y is holomorphic on W \ P . Note that in W \ P , Y has only isolated zeroes.
2) Since P is the exceptional divisor of g, h(P ) is the exceptional divisor of s and isF -invariant. Then is a regular covering map.
3) Let C θ i be the trajectory of Z 0 through θ i . Y is a complete holomorphic vector field on W \ {g −1 (C θ i )} s i=1 . Each g −1 (C θ i ) \ P is contained in a trajectory R z i of Y . Let us fix one of them, say C θ j . Let us set Θ ∩ C θ j = {θ j l } h l=0 taking j 0 = j. Note that R z j l = R z j for any l. For every θ j l there is a pointθ j l ∈ P such that R z j ∪ {θ j l } defines a separatrix ofF throughθ j l . Note thatθ j l is the unique singular point ofF in P such that g(θ j l ) = θ j l . We can take around θ j l a neighbourhood U and coordinates (z, w) such thatF is generated by z∂/∂z − w∂/∂w where (R z j ∪ {θ j l }) ∩ U = {w = 0} and g −1 (θ j l ) ∩ U = {z = 0}. As Y has a pole of order one along {z = 0}, it follows that is not complete. However, it extends on R z j ∪ {θ j l } h l=0 as a complete vector field because g restricted to R z j extends to R z j ∪ {θ j l } h l=0 as a biholomorphism onto C θ j and Global holomorphic vector fields [2]. The list of holomorphic vector fields with isolated singularities on compact complex surfaces is well known. In [2, Chapter 6] we can find the details when the surface is projective. In particular, for Z 0 on S we have one of the following possibilities: I) S has an elliptic fibration f : S → B, and Z 0 is a nontrivial holomorphic vector field on S tangent to the fibres of f . Each fibre of f is a smooth elliptic curve which can be multiple, and outside multiple fibers f is a locally trivial fibration. Moreover Z 0 has empty zero set. II) S = C 2 /Λ is a 2-torus and Z 0 is a linear vector field on it, that is, the quotient of a constant vector field on C 2 . III) S is a CP 1 -bundle over an elliptic curve E, and Z 0 is transverse to the fibers and projects on E to a constant vector field. In this case Z 0 is the suspension of E via the representation ρ : π 1 (E) → Aut(CP 1 ) associated to the bundle structure, and it generates a Riccati foliation without invariant fibres and whose monodromy map is ρ. IV) S is a rational surface, and up to a birational map we have Y = CP 1 × CP 1 and Z 0 = v 1 ⊕ v 2 , where v 1 and v 2 are holomorphic vector fields on CP 1 . In the course of the proof we will consider S in some cases as a differentiable manifold with a given Riemannian metric g. If (N, g) is a Riemannian manifold, we denote by d the distance given by the metric, and by B d r (p) the open ball centered at p. For the basic notions of Riemannian geometry that we will be used in the rest of the paper, see [16].
We will analyze the possible cases for Z 0 and S. First note that Case I) does not really occur since F has not rational first integral.

Cases II) and III).
Proposition 1. If Z 0 and S are as in II) or III) any irreducible component of D ∪ E is invariant byF .
Proof. Let D 0 be an irreducible component of D ∪ E that is not invariant bỹ F. There is a compact curve Q 0 (possibly singular) in S generically transversal to Z 0 . It is enough to define Q 0 as one of the connected components of r −1 (s(D 0 )). Note that s(D 0 ) is not a point.
Case II). Let us take S as the quotient manifold C 2 /Λ. We identify C 2 with R 4 , and Λ is an integral lattice of rank four. Remark 6. Let µ : C 2 → C 2 /Λ denote the canonical submersion map. If we consider R 4 with the usual euclidean g, taking g ′ = µ * g as the metric on C 2 /Λ, the map µ becomes a Riemannian covering map. We will denote by d and d ′ the distances in (R 4 , g) and (R 4 /Λ, g ′ ), respectively.
The vector field Z 0 is the projection by µ of a constant vector field on C 2 , and thus its trajectories must be of the form µ(L t ) where {L t } t∈C is the family of lines parallel to a given direction. Note that Z 0 has not singularities.
Lemma 3. There is a compact K S such that Q 0 ⊂K Proof. Since Z 0 is complete and without singularities we can define where ϕ : C × S → S is the complex flow of Z 0 . If we apply the Flow Box Theorem to the points of Q 0 we easily deduce that Q 0 ⊂K.
We will use that α has the following property with respect to K. If α(t) = 0, we suppose µ(L t ) ∩K = ∅ and obtain a contradiction.
Otherwise we can determine a sequence {x n } n∈N ⊂ Q 0 converging to x * ∈ Q 0 and such that d ′ (µ(L t ), x n ) < 1/n because Q 0 is compact and (R 4 /Λ, g ′ ) is complete. But it implies that for any ball B d ′ r (x * ) there exists n(r) ∈ N + such that By contradiction, suppose that d(L t , µ −1 (Q 0 )) < δ. Then there are z ∈ L t andz ∈ µ −1 (Q 0 ) with d(z,z) < δ. Note that µ −1 (Q 0 ) is an analytic variety (non necessarily compact) of C 2 and that µ(z) = µ(z) by Fact 1. Let c be a segment from z toz. As µ defines a local isometry from (R 4 , g) to (R 4 /Λ, g ′ ), we can take B d r i (z i ) ⊂ R 4 , 0 ≤ i ≤ s, centered at z i ∈ c, where z 0 = z and z s =z, and in such a way that µ restricted to each B d r i (z i ) defines an isometry over its image. Moreover, we can assume that B d r i (z i ) ∩ B d r j (z j ) = ∅ if and only if j = i + 1, and thus fix s − 1 points z i,i+1 in these intersections. As the isometries preserve intrinsic distance, Fact 1 and the triangle inequality implies the following contradiction Lemma 5. µ(L t ) ∩ Q 0 = ∅ for any t ∈ C, and then α ≡ 0.
It follows from Remark 4 that M i is contained in a trajectory of Z 0 . Hence there is L s i such that µ(L s i ) ⊃ M i .
In particular, µ(L s i ) and M i are respectively biholomorphic to C and C * .
Proof. Lemma 5 implies that µ(L s i ) ∩ Q 0 = ∅. Moreover it is clear that µ(L s i ) ∩ Q 0 ⊂ µ(L s i ) \ M i . It follows from Remark 4 that r * (s * (X |Cz )) |M i is complete and then it extends holomorphically by zeroes on µ(L s i )\M i . Since C z is not algebraic, M i is biholomorphic to C * and µ(L s i )∩Q 0 = µ(L s i )\M i is an unique point p i . Thus µ(L s i ) must be biholomorphic to C.
Since the foliation defined by Z 0 on S has codimension 1 and it has not singularities, the closure of µ(L s i ) in the open set U ′ ⊂ S of non-compact leaves, that we will denote by L ′ , is a subvariety of real codimension 0, 1 or 2 [9, Théorème 1.4]. It holds U ′ = S and then L ′ is the closure of µ(L s i ) in S. If there were one compact leaf J, [9, Théorème 1.4] also assures that any non-compact leaf must accumulate J. In particular µ(L s i ) accumulates J. On the other side, as Q 0 cut any leaf (Lemma 5), it must cut J, and µ(L s i ) accumulates the points of J ∩ Q 0 , which is not possible since p i is the unique point in µ(L s i ) \ M i (Lemma 6). Note that L ′ is S or a real compact subvariety of dimension three. If L ′ had real codimension 2, it would define a real compact subvariety of dimension two of S (L ′ is closed in S) containing µ(L s i ), which is a non-algebraic leaf. One concludes that µ(L s i ) must intersect infinitely many times Q 0 , and then one obtains again a contradiction with Lemma 6.
Case III). Let us consider S as CP 1 -bundle over an elliptic curve E with bundle projection p : S → E. The structure of S can be lifted as CP 1 -bundlẽ S over C via the universal covering map Γ : C → E: we can determine a complex surfaceS, a holomorphic covering F :S → S and a bundle projectionp :S → C such that p • F = Γ •p.
Moreover as C is contractible this CP 1 -bundle is trivial. ThusS = C × CP 1 andp(x, y) = x is the projection over the first factor. Lemma 7. There is a holomorphic automorphism σ of C × CP 1 such that σ * (F * Z 0 ) is the horizontal vector field.
Proof. It is clear that F * Z 0 generates a Riccati foliation adapted top and without invariant fibres. If σ(t, y) =φ(t, 0, y), withφ the complex flow of F * Z 0 , σ is bijective, since each trajectory of F * Z 0 intersects each fibre of p in only one point, and σ(C × {y}) are the trajectories of F * Z 0 . After Lemma 7, the trajectories of Z 0 are of the form (F • σ)(L t ) where {L t } t∈CP 1 is now the family of lines L t = C × {t}.
Remark 8. As S is compact, S (as real manifold) admits a Riemannian metric g ′ . Let us setμ = F • σ. The mapμ from (R 2 × S 2 ,μ * g ′ ) to (S, g ′ ) is a local Riemannian isometry. But still more, F is a covering map and σ is a biholomorphism, hence F • σ is also a covering map andμ is a Riemannian covering map. As (S, g ′ ) is compact, it is complete, and (R 2 × S 2 ,μ * g ′ ) is complete. We will denote by d and d ′ the distances in (R 2 × S 2 ,μ * g ′ ) and (S, g ′ ), respectively.
The vector field Z 0 is complete and without zeroes. We will consider as in case II) an irreducible component D 0 of D ∪ E that is not invariant byF, and the compact curve Q 0 (possibly singular) in S generically transversal to Z 0 , defined by one of the connected components of r −1 (s(D 0 )). As in Lemma 3 we can determine a compact set K S such that Q 0 ⊂K.
We will consider the continuous map (it follows as in Remark 7) Once we have fixedμ = F • σ, the complete metrics in Remark 8, the compact set K and the mapᾱ as (8), we can prove similar Lemmas to Lemmas 3, 4, 5 and 6 of Case III), where µ and α must be substituted byμ andᾱ in their statements.
Let L s i be such thatμ(L s i ) ⊃ M i . Since p |μ(Ls i ) :μ(L s i ) → E is a covering map, andμ(L s i ) is biholomorphic to C (Lemma 6),μ(L s i ) must cut almost all the fibres of p infinitely many times. Let κ ∈ E such that p −1 (κ) contains an infinite sequence of different points in p −1 (κ) ∩μ(L s i ). By compactness of p −1 (κ), the above sequence converges to q 1 ∈ p −1 (κ). Note that q 1 is a regular point of Z 0 . Ifμ(Ls) is the trajectory of Z 0 through q 1 ,μ(L s i ) must accumulateμ(Ls) (flow-box theorem). On the other hand, µ(Ls) ∩ Q 0 = ∅ (Lemma 5) implies a contradiction with the fact that p i is the unique point inμ(L s i ) \ M i (Lemma 6).
Remark 9. One can also obtain a contradiction by distinguishing several cases, according to the (abelian) monodromy Γ ⊂ Aut(CP 1 ). If Γ has rank 1, then the non-algebraic leaves of Z 0 are isomorphic to C * , and one gets a contradiction by using Remark 5, and the fact that the intersection with algebraic curves is nonempty. If Γ has rank 2 then Γ =< f, g > with f (z) = αz, g(z) = βz or f (z) = z + 1, g(z) = z + w. In the first case (where, moreover, α n β m = 1 for every (m, n) = (0, 0)) the non-algebraic leaves are sufficiently dense to apply the same argument as in case II. In the second case one can prove that every algebraic curve C ⊂ S different from the elliptic curve E = {z = ∞} must intersect E, and from this fact it follows again that every non-algebraic trajectory of Z 0 intersects C infinitely many times.

Case IV). There is a birational transformation
where v 1 and v 2 are holomorphic vector fields on CP 1 . The description of G can be found in [2, p. 87]. In particular, G is a finite sequence of birational transformations which are contractions of curves invariant by Z 0 or blowing-ups at zeros of Z 0 . Hence the exceptional divisor of G does not meet M i , and as consequence G(M i ) is biholomorphic to M i . But still more, as M i supports a complete vector field according to Remark 4, we can define an entire curve f : C → G(M i ). In absence of rational first integrals, we may assume that G * Z 0 is not constant. Note that G(M i ) is contained in a trajectory L z of G * Z 0 and that L z \ G(M i ) is empty or one point. There are two cases for G * Z 0 : a) v 1 and v 2 with zeroes of order one at 0 (λz∂/∂z + µw∂/∂w). b) v 1 with a zero of order one at 0 and v 2 constant (λz∂/∂z + µ∂/∂w).
Let CP 2 be the compactification of B. The image off : C → G(M i ) ∩ B is contained in CP 2 minus at least four hypersurface sections, that is, three sections defined by the algebraic curves of Lemma 8 along with the line at infinity CP 2 \ B. According to Green's Theorem [11, p. 199],f (C) must be contained in some algebraic curve, what contradicts our assumptions. Hence 1), 2) and 3) of the statement of Proposition follows.
Note that C z ,C z ,Ĉ z , G(M i ) and M i are biholomorphic to C * , and that L z ≃ C and G(M i ) = L z \ {q} ≃ C * with q ∈ {w = 0} \ {(0, 0), (∞, 0)}. G(M i ) has two parabolic ends, which are properly embedded in the complementary set of {z = ∞} ∪ {w = ∞} in CP 1 × CP 1 . One Σ 1 defined by a punctured disk centered at q, that is algebraic; and other Σ 2 defined by G(M i ) \ Σ 1 , that is transcendental and accumulates {w = ∞}. Note that G * Z 0 has two saddle-nodes as singularities. One at (0, ∞), with strong separatrix inside {w = ∞} and weak separatrix inside {z = 0}; and other one at (∞, ∞), with strong separatrix inside {w = ∞} and weak separatrix inside {z = ∞}. On the other hand G * Z 0 defines a Riccati foliation adapted to β(z, w) = w. One may assume (maybe after blowing-up reduced singularities) thatF is Riccati with respect to β W = β • G • g and that G • g is the contraction of curves inside fibers of β W that produces the local models of [4, p. 439]. In this case all the fibers {w = c}, with c = ∞, are transversal minus one that is tangent, {w = ∞}, and of class (d).
Since h is an algebraic covering map from W to M , the proper mapping theorem allows to define the trace of β W as a rational function β M on M [10]. Moreover, one can assume that β M is a fibration after eliminating its base points. Recall that the property of being reduced is stable by blowing ups. Moreover, the possible dicritical components of the resolution of the pencil given by β M must be transversal to the corresponding foliation.
By construction, the generic fiber F of β M is a curve transverse toF. Note that D 0 must be contained in a fiber F 0 of β M as a consequence of 3) in the statement of this Proposition. Let us consider the following cases according to the genus of F . • If F is of genus ≥ 2, it follows from [15, Theorem III.6.1] thatF has a rational first integral, which is not possible.
• If F is of genus 1,F is a Turbulent foliation. Let us see that this case neither occurs because it would imply the existence of a rational first integral as before. Indeed, note that F does not cut F 0 since β M is a fibration. On the other hand, D ∩ F = ∅ by the maximum principle. As D 0 is the unique irreducible component of D ∪ E that is not invariant byF, there must be oneF-invariant component D 2 of D such that D 2 ∩ F = ∅. The existence of D 2 implies thatF has a first integral (Lemma 1).
• If F is of genus 0,F is a Riccati foliation. Let us see thatF satisfies 4) of the statement of Proposition 2. After contraction of rational curves each fiber of β M admits one of the five possible models in [3, § 7], [4, p. 439]. If there is one fiber F 1 tangent toF, as D 0 is the unique irreducible component of D ∪ E which is not invariant byF and it is contained in F 0 , then F must cut D in only one or two points near F 0 . Then we can conclude as in Lemma 2 that β M projects by π as a rational function R of type C or C * . Finally, one shows that all the fibers of β M are not transverse to F . In the contrary case, if L 0 is the leaf defined byC z , as the covering map β M |L : L → CP 1 is not finite (otherwise L is compact and C z is algebraic), L must cut infinitely many times F 0 and C z is not isomorphic to C * , which is not possible.
After Proposition 2 we can assume that any irreducible component of D ∪ E is invariant byF . Otherwise Theorem 1 follows by the results of §2.

3.3.
Existence of a second integral. Let us come back to the beginning of §3, and consider (4). Proof. Let R 0 be a connected component of h −1 (C z ). As R 0 does not meet the exceptional divisor of s : M →M then h |R 0 : R 0 →C z is a nonramified finite covering map. Hence h * |R 0 (X |Cz ) =X |R 0 is complete. On the other handX and Y are tangent on R 0 according to (4). Thus R 0 is a Riemann Surface contained in a trajectory R z of Y . Let ϕ z : Ω z → R z be the corresponding solution. We have two possibilities from 3.-of Remark 5: Let us suppose that R z = R z j . We take the solution f z j : C → R z j ∪ {θ j l } h l=0 of (5) and the discrete subset ∆ = {f z j −1 (θ j l )} h l=0 of C. Since f z j |C\∆ = ϕ z then Ω z = C \ ∆. It follows from i) and ii) that ϕ z is a univaluated holomorphic map. Let us note thatX |Rz must be also complete becauseX |R 0 is complete. Using these two facts and that T ∈ Ω z → ϕ z (T ) ∈ R z is a covering map then (9) ϕ * z (X |Rz ) = ϕ * z (X)(T ) = (F • ϕ z (T )) · ϕ * z (Y ) = (F • ϕ z (T )) ∂ ∂T is a complete vector field on Ω z . What is only possible if Ω z = C or C * and (F • ϕ z )(T ) = aT + b, for a, b ∈ C. We conclude that Y (F )(ϕ z (T )) = (F • ϕ z ) ′ (T ) is constant and hence Y 2 F vanishes along R z , which can be assumed to be non-algebraic since C z is by hypothesis. Hence Y 2 F = 0. Let us take a point z ∈ W ′ = W \ ({F = 0} ∪ {F = ∞} ∪ P ). If S z is the trajectory ofX through z, as Y is holomorphic on W ′ (1.-of Remark 5) and tangent toX on S z by (4) then S z defines a trajectory R z of Y . Since it holds (9), the fact that Y 2 F = 0 implies thatX |Sz is complete.