The structure and physical significance of the full set of solutions to coupled-cluster (CC) equations at various stages of the dissociation process and the impact of the choice of reference functions on these solutions have been studied for the first time. The equations for the CC method involving double excitations (CCD) are obtained for the P4 model consisting of two H-2 molecules in a rectangular nuclear configuration determined by a geometry parameter alpha. We consider equations for the reference states \Phi(A)>, \Phi(Q)>, and \Phi(B)> corresponding to the lowest, highest, and intermediate Hartree-Fock (HF) energies, respectively. The first two states provide a size-consistent description of the dissociation process. For the compact-molecule geometries (alpha < 10.0) the sets of complete solutions to the standard CCD equations [based on molecular orbitals (MOs) of D-2h symmetry] in the spin-orbital and spin-symmetry-adapted versions always consist of 20 and 12 entries, respectively. For \Phi(A)> and \Phi(B)> in the dissociation limit (alpha -->infinity) only for the latter version the solutions can be attained by homotopy method. In this case we have reformulated the standard spin-symmetry-adapted CCD equations to a version based on the use of localized orbitals (LO) which is extremely simple and can be solved analytically providing an understanding of the unexpected peculiarities of the solutions for alpha -->infinity. For \Phi(A)> and \Phi(Q)>, there are only two regular solutions. For the remaining 10 solutions, the CCD wave functions are meaningless despite the fact that the corresponding CCD energies are equal to the exact values.