On integral conditions for the existence of first integrals analytic saddle singularities
We study one-parameter analytic integrable deformations of the germ of $2\times(n-2)$-type complex saddle singularity given by $d(xy)=0$ at the origin $0 \in \mathbb C^2\times \mathbb C^{n-2}$. 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 $\omega_j$ are holomorphic one-forms in some neighborhood of the origin $0\in \mathbb C^n$. We prove that, under a nondegeneracy condition of the singular set of the deformation, with respect to the fibration $d(xy)=0$, the existence of a holomorphic first integral for each element ${\omega}^t$ 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 $\gamma_c$ 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 $x^ny^m=c$ 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 analytic center-cylinder type singularities in terms of the vanishing of some easy to compute line integrals.