  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.
