Systematic construction of nonautonomous Hamiltonian equations of Painleve type. I. Frobenius integrability

This article is the first one in a suite of three articles exploring connections between dynamical systems of Stackel type and of Painleve type. In this article, we present a deformation of autonomous Stackel-type systems to nonautonomous Frobenius integrable systems. First, we consider quasi-Stackel systems with quadratic in momenta Hamiltonians containing separable potentials with time-dependent coefficients, and then, we present a procedure of deforming these equations to nonautonomous Frobenius integrable systems. Then, we present a procedure of deforming quasi-Stackel systems with so-called magnetic separable potentials to nonautonomous Frobenius integrable systems. We also provide a complete list of all two- and three-dimensional Frobenius integrable systems, both with ordinary and with magnetic potentials, which originate in our construction. Further, we prove the equivalence between both classes of systems. Finally, we show how Painleve equations PI-PIV can be derived from our scheme.Funding Agencies|Polish Ministry of Science and Higher EducationMinistry of Science and Higher Education, Poland