A class of parameter-dependent commuting matrices

We present a novel class of real symmetric matrices in arbitrary dimension d, linearly dependent on a parameter x. The matrix elements satisfy a set of nontrivial constraints that arise from asking for commutation of pairs of such matrices for all x, and an intuitive sufficiency condition for the solvability of certain linear equations that arise therefrom. This class of matrices generically violates the Wigner von Neumann non-crossing rule, and is argued to be intimately connected with finite-dimensional Hamiltonians of quantum integrable systems.