The Maslov index and nondegenerate singularities of integrable systems

We consider integrable Hamiltonian systems in R-2n with integrals of motion F = (F-1,..., F-n) in involution. Nondegenerate singularities of corank one are critical points of F where rank d F = n - 1 and which have definite linear stability. The set of corank-one nondegenerate singularities is a codimension-two symplectic submanifold invariant under the flow. We show that the Maslov index of a closed curve is a sum of contributions 2 from the nondegenerate singularities it encloses, the sign depending on the local orientation and stability at the singularities. For one-freedom systems this corresponds to the well-known formula for the Poincare index of a closed curve as the oriented difference between the number of elliptic and hyperbolic fixed points enclosed. We also obtain a formula for the Liapunov exponent of invariant (n - 1)-dimensional tori in the nondegenerate singular set. Examples include rotationally symmetric n-freedom Hamiltonians, while an application to the periodic Toda chain is described in a companion paper (Foxman and Robbins 2005 Nonlinearity 18 2795-813).