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This article is the first one in a suite of three articles exploring connections between dynamical systems of Stäckel type and of Painlevé type. In this article, we present a deformation of autonomous Stäckel‐type systems to nonautonomous Frobenius integrable systems. First, we consider quasi‐Stäckel 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‐Stäckel 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 Painlevé equationsP  I   −   P   I  V$P_{I}-P_{IV}$    can be derived from our scheme.

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