The, established by Galois, relationship between the roots of the modular equation, of level p, and the p-torsion points of a corresponding elliptic curve serves as a basis of exact and fast algorithms for solving modular equations of all degrees! The constructive approach of Galois turns out being fruitful in developing efficient algorithms for evaluating exact solutions of fundamental problems of mechanics .  An indispensable condition for such an algebraic approach is the identification of the complete symmetry group of the solution and the establishment of all (without exception) transformations amongst solutions. Concrete and clear examples will be provided to demonstrate the actual necessity for such a complete and constructive study.