  This paper is a continuation of our previous work "Six-vertex model and
non-linear differential equations I. Spectral problem" in which we have put
forward a method for studying the spectrum of the six-vertex model based on
non-linear differential equations. Here we intend to elaborate on that approach
and also discuss properties of the spectrum unveiled by the aforementioned
differential formulation of the transfer matrix's eigenvalue problem. In
particular, we intend to demonstrate how this differential approach allows one
to study continuous symmetries of the transfer matrix's spectrum through the
Lie groups method.
