  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.
