On Differential Equations of Integrable Billiard Tables

We introduce a method to find differential equations for functions which define tables, such that associated billiard systems admit a local first integral. We illustrate this method in three situations: the case of (locally) integrable wire billiards, for finding surfaces in ℝ³ with a first integral of degree one in velocities, and for finding a piece-wise smooth surface in ℝ³ homeomorphic to a torus, being a table of a billiard admitting two additional first integrals.