  In this paper, we show that there is a large class of fermionic systems for
which it is possible to find, for any dimension, a finite closed set of
eigenoperators and eigenvalues of the Hamiltonian. Then, the hierarchy of the
equations of motion closes and analytical expressions for the Green's functions
are obtained in terms of a finite number of parameters, to be self-consistently
determined. Several examples are given. In particular, for these examples it is
shown that in the one-dimensional case it is possible to derive by means of
algebraic constraints a set of equations which allow us to determine the
self-consistent parameters and to obtain a complete exact solution.
