Two-dimensional integrable potentials with quartic invariants

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Abstract
For autonomous two-dimensional conservative dynamical systems the author derives four necessary and sufficient conditions which the potential function U(x, y) has to satisfy in order that it is integrable with the second constant of motion quartic in the velocity components. The author also develops the method by means of which he finds the quartic invariant for a given potential satisfying these conditions. Certain degenerate cases leading to pseudo-quartic integrals are discussed. Two examples and a counter-example are presented.