On applications of Maupertuis-Jacobi correspondence for Hamiltonians F(x,|p|) in some 2-D stationary semiclassical problems

We make use of the Maupertuis -- Jacobi correspondence, well known in Classical Mechanics, to simplify 2-D asymptotic formulas based on Maslov's canonical operator, when constructing Lagrangian manifolds invariant with respect to phase flows for Hamiltonians of the form $F(x,|p|)$. As examples we consider Hamiltonians coming from the Schr\"odinger equation, the 2-D Dirac equation for graphene and linear water wave theory.