Systematic construction of non-autonomous Hamiltonian equations of Painlev\'{e}-type. I. Frobenius integrability

This article is the first one in a suite of three articles exploring connections between dynamical systems of St"{a}ckel-type and of Painlev'{e}- type. In this article we present a deformation of autonomous St"{a}ckel-type systems to non-autonomous Frobenius integrable systems. First, we consider quasi-St"{a}ckel systems with quadratic in momenta Hamiltonians containing separable potentials with time dependent coefficients and then we present a procedure of deforming these equations to non-autonomous Frobenius integrable systems. Then, we present a procedure of deforming quasi-St"{a}ckel systems with so called magnetic separable potentials to non-autonomous Frobenius integrable systems. We also provide a complete list of all $2$- and $3$-dimensional Frobenius integrable systems, both with ordinary and with magnetic potentials, that originate in our construction. Further, we prove the equivalence between both classes of systems. Finally we show how Painlev'{e} equations $P_I-P_{IV}$ can be derived from our scheme.