A Galois–Eisenstein ring or a GE-ring is a finite commutative chain ring. We consider two methods of enumeration of all solutions of some system of polynomial equations over a GE-ring R. The first method is the general method of coordinate-wise linearisation. This method reduces to solving the initial polynomial system over the quotient field = R/ Rad R and then to solving a series of linear equations systems over the same field. For an arbitrary ideal of the ring R[x ₁, . . . , x _k ] a standard base called the canonical generating system (CGS) is constructed. The second method consists of finding a CGS of the ideal generated by the polynomials forming the left-hand side of the initial system of equations and solving instead of the initial system the system with polynomials of the CGS in the left-hand side. For systems of such type a modification of the coordinate-wise linearisation method is presented.