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The n-particle periodic Toda chain is a well known example of an integrablebut nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-foldsingularities of the Toda chain, ie points where there exist k independentlinear relations amongst the gradients of the integrals of motion, coincidewith points where there are k (doubly) degenerate eigenvalues ofrepresentatives L and Lbar of the two inequivalent classes of Lax matrices(corresponding to degenerate periodic or antiperiodic solutions of theassociated second-order difference equation). The singularities are shown to benondegenerate, so that Sigma_k is a codimension-2k symplectic submanifold.Sigma_k is shown to be of elliptic type, and the frequencies of transverseoscillations under Hamiltonians which fix Sigma_k are computed in terms ofspectral data of the Lax matrices. If mu(C) is the (even) Maslov index of aclosed curve C in the regular component of R^{2n}, then (-1)^{\mu(C)/2} isgiven by the product of the holonomies (equal to +/- 1) of the even- (or odd-)indexed eigenvector bundles of L and Lmat.Comment: 25 pages; published versio

Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain