Similar to their classical counterpart, obtaining such symmetries is based on solving the so called "determining equations" (DE) for the infinitesimal symmetries. In the classicalcase, these equtions form the overdetermined linear systems of PDEs, and the degree of their redundancy permits one to easily solve them. In contrast to Lie's symmetries, the DE for conditional symmetries are nonlinear and less overdetermined. This explains that as a rule only particular solutions of nonclassical DE were obtained even in the case of PDEs in two independent variables. The situation was essentially improved after several authors had developed computer algebras based algorithms for reducing the overdetermined systems to passive, or involutive forms.
But even with such programs, it takes too much time and sometimes too many computer resources to get the general solution of nonclassical DE. Here two ways out are possible: either to restrict oneself to particular but well determined classes of conditional symmetries or to develop more efficient algorithms and packages.One of the most attractive ways in the spirit of symmetry theory is to restrict the family of conditional symmetries by taking into account the symmetry properties of the conditional symmetries w.r.t. the classical ones. This allows to get conditional symmetries with an additional advantage that the quotient equations for the invariant solutions inherit the classical symmetries. After conditional symmetries are obtained, the problem of solving the quotient equations (QE) appears. In the case of PDEs in two independent variables, the QE are ODEs but in general they are not explicitly solvable. Here we again can take advantage of computer algebra systems supplied with the means for numerical solutionof ODEs and graphical representation of the approximate solutions.
All the considerations made above are illustrated with the author's results in symmetry research with the ``Mathematica'' program SYMMAN.
