On Integral Conditions for the Existence of First Integrals for Analytic Deformations of Complex Saddle Singularities

Abstract

We study one-parameter analytic integrable deformations of the germ of 2 × (n − 2)-type complex saddle singularity given by d(xy) = 0 at the origin $0 \in \mathbb {C}^{2}\times \mathbb {C}^{n-2}, n \geq 2$. Such a deformation writes ${\omega }^{t}=d(xy) + \sum \limits _{j=1}^{\infty } t^{j} \omega _{j}$ where $t\in (\mathbb {C},0)$ is the parameter of the deformation and the coefficients ω_j are holomorphic one-forms in some neighborhood of the origin $0\in \mathbb {C}^{n}$. We consider a natural condition on the singular set of the deformation with respect to the fibration d(xy) = 0. Under this condition, the existence of a holomorphic first integral for each element ω^t of the deformation is equivalent to the vanishing of certain line integrals $\oint _{\gamma _{c}}{\omega }^{t}=0, \forall \gamma _{c}, \forall t$ calculated on cycles γ_c contained in the fibers $(xy=c), 0 \ne c \in (\mathbb {C},0)$. This result is quite sharp regarding the conditions of the singular set and on the vanishing of the integrals in cycles. It is also not valid for ramified saddles, i.e., for deformations of saddles of the form xⁿy^m = c where n + m > 2. As an application of our techniques we obtain a criteria for the existence of first integrals for integrable codimension one deformations of quadratic real analytic center-cylinder type singularities in terms of the vanishing of some easy to compute line integrals.