  We consider integrable Hamiltonian systems in R^{2n} with integrals of motion
F = (F_1,...,F_n) in involution. Nondegenerate singularities are critical
points of F where rank dF = n-1 and which have definite linear stability. The
set of 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 is 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
Poincar\'e 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.
