Mean-field calculations in multidimensional deformation spaces are performed and the shape coexistence and isomers generated by exotic nuclear configurations and toroidal and superdeformed ones are addressed. We use a phenomenological mean-field Hamiltonian of Woods-Saxon type with its universal parametrization involving eight parameters fixed once for all for the full periodic table. Original parametric correlations existing in this type of Hamiltonians are removed using methods of inverse problem theory of applied mathematics. Stochastic analysis of uncertainties of the final nuclear energy predictions with the obtained correlation-free parametrization is performed and the viability tests are illustrated and discussed. Prediction capacities of resulting model related to the description of the nuclear shape properties are cross-checked using the experimental information available, revealing full coherence. Presented results encourage experimental verification of predicted exotic structures; suggestions related to identification possibilities are formulated and discussed.