  Using recently proposed method of discrete Hirota dynamics for integrable
(1+1)D quantum field theories on a finite space circle of length L, we derive
and test numerically a finite system of nonlinear integral equations for the
exact spectrum of energies of SU(N)xSU(N) principal chiral field model as
functions of m L, where m is the mass scale. We propose a determinant solution
of the underlying Y-system, or Hirota equation, in terms of determinants
(Wronskians) of NxN matrices parameterized by N-1 functions of the spectral
parameter, with the known analytical properties at finite L. Although the
method works in principle for any state, the explicit equations are written for
states in the U(1) sector only. For N>2, we encounter and clarify a few
subtleties in these equations related to the presence of bound states, absent
in the previously considered N=2 case. In particular, we solve these equations
numerically for a few low-lying states for N=3 in a wide range of m L.
