  We revisit the symplectic aspects of the spectral transform for matrix-valued
rational functions with simple poles. We construct eigenvectors of such
matrices in terms of the Szeg\H{o} kernel on the spectral curve. Using
variational formulas for the Szeg\H{o} kernel we construct a new system of
action-angle variables for the canonical symplectic form on the space of such
functions. Comparison with previously known action-angle variables shows that
the vector of Riemann constants is the gradient of some function on the moduli
space of spectral curves; this function is found in the case of matrix
dimension 2, when the spectral curve is hyperelliptic.
