On differential equations of integrable billiard tables

The paper proves that (a) the wire billiard defined by γ(t)=e^{At}γ0 with A∈so(n) has a linear first integral F=∑_{i<j}a_{ij}(vjxi−vixj), and (b) the surface z=−(β/α)Arctan(x/y)+f(x^2+y^2) admits the degree-one billiard integral F=α(xvy−yvx)+βvz. The model independently verifies both results via direct invariance checks: along free flight (v constant) and at reflections (angle preservation for wires; tangential component preserved for surfaces with a tangent Killing-type field). The arguments align with Theorem 1 and Theorem 2 in the paper, but use a more geometric verification instead of the paper’s identity/PDE method, hence both correct with different proofs .