  The study of the recently constructed group foliation for the geopotential
forecast equation is continued. The group foliation consists of two systems,
namely the automorphic and resolving systems, the analysis of which facilitates
the derivation of invariant solutions for the original equation. As obtaining a
general solution to the resolving system (even to its reductions on subgroups)
is problematic, its various particular solutions are considered. Consequently,
the question arises concerning the specific forms of automorphic systems that
correspond to exact solutions obtained through alternative methods. This is of
interest for both comparing solutions derived through different approaches and
for the integration of specific automorphic systems. The problem is discussed
in a number of examples.
