Another billiard problem

In the works of S.Y.Dobrokhotov and V.E.Nazaikinsky on the semiclassical approximation for the degenerate wave equation, the problem of studying the geodesic flow of a Riemannian metric tending to infinity on the boundary of the domain arose. After regularization, the problem is reduced to the study of a degenerate Birkhoff-type billiard. A normal form of the regularized geodesic flow near the boundary will be obtained and a version of Lazutkin's theorem for the corresponding billiard map will be proved.