The impact of approximations to the form of the cluster operator on the structure and physical significance of the complete set of geometrically isolated solutions to the coupled-cluster (CC) equations has been studied for the first time. To systematically study the correspondence of solutions obtained at various levels of the approximation process, a continuation procedure based on a set of beta-nested equations (beta-NE) has been proposed and applied. Numerical studies based on a homotopy method for obtaining full solutions to sets of polynomial equations have been performed for the H4 and P4 models which belong to the simplest realistic many-electron model systems. Two examples of approximation procedures have been considered. The first one involved, for the P4 model, the approximation leading from the full CC (FCC) method to the CC method based on double excitations (CCD). As a result of this approximations the number of solutions has increased from 8 to 20. In the second example, for H4, we have studied the approximation leading from the CCSD method to the CCD one. To complete these studies, we have for the first time obtained the full set of geometrically isolated solutions for a CCSD equations which consists of 60 solutions. Only a small subset of this set might have some physical significance. During the approximation process considered, the number of solution decreases from 60 to 12. This radical drop of the numbers of solutions is a consequence of the absence of the third and fourth powers of the unknowns in the CCD equations.