Small C¹-smooth perturbations of skew products and the partial integrability property

In this paper we investigate stability of the integrability property of skew products of interval maps under small C¹-smooth perturbations satisfying some conditions. We obtain here (sufficient) conditions of the partial integrability for maps under considerations. These conditions are formulated in the terms of properties of the unperturbed skew product. We give also the example of the partially integrable map.