New integrable systems from unitary matrix model

We show that the one-dimensional unitary matrix model with potential of the form aU+bU2+h.c. is integrable. By reduction to the dynamics of the eigenvalues, we establish the integrability of a system of particles in one space dimension in an external potential of the form a cos(χ+α)+b cos(2χ+β) and interacting through two-body potentials of the inverse sine square type. This system constitutes a generalization of the Sutherland model in the presence of external potentials. The positive-definite matrix model, obtained by analytic continuation, is also integrable, which leads to the integrability of a system of particles in hyperbolic potentials interacting through two-body potentials of the inverse hyperbolic sine square type. (orig.)