To establish the existence and origin of the nonalgebraic irregularities of solutions to coupled-cluster (CC) equations and to indicate ways of their elimination, we have revisited the two analytically solvable characteristic equations (CE) studied by Zivkovic and Monkhorst [J. Math. Phys. 19, 1007 (1978)]. The results of these studies have strongly influenced the general conclusions concerning the possible types of singularities. We present some arguments that the most serious irregularities-the nonnormal and resonance ones-are a result of the special structures of the CEs considered. The CE employed for the demonstration of resonance solutions is not physically representable, which raises the hope that such solutions will not appear in quantum-chemical applications of the coupled-cluster method. It is proved that the presence of nonnormal solutions is a consequence of the existence of such passive diagonal blocks of the Hamiltonian matrix which share a common eigenvalue. Such blocks can be eliminated by taking into account the symmetry species of the basis functions involved, which is most effectively done by proceeding to a symmetry-adapted formulations. Therefore, one may eliminate or at least reduce the number of nonnormal solutions to the CC equations by proceeding to their symmetry-adapted versions.