  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.
