Differential Invariants, Hidden and Conditional Symmetry.

We present a survey of development of the concept of hidden symmetry in the field of partial differential equations, including a series of results previously obtained by the author. We also add new examples of the classes of equations with hidden symmetry of type II and explain the nature of the earlier established nonclassical symmetry of some equations. We suggest a constructive algorithm for the description of the classes of equations, which have specified conditional or hidden symmetry and/or can be reduced to equations with smaller number of independent variables by using a specific ansatz. We consider reductions existing due to the presence of Lie and conditional symmetry and also of the hidden symmetry of type II. We also discuss relationships between the concepts of hidden and conditional symmetry. It is shown that the hidden symmetry of type II earlier regarded as a separate type of non-Lie symmetry is caused, in fact, by the nontrivial Q-conditional symmetry of the reduced equations. The proposed approach enables us not only to find hidden symmetry and new reductions of the well-known equations but also to describe a general form of equations with given Q-conditional and type-II hidden symmetry. As an example, we describe the general classes of equations with hidden and conditional symmetry under rotations in the Lorentz and Euclid groups for which the corresponding hidden and conditional symmetry allows their reduction to radial equations with smaller number of independent variables. [ABSTRACT FROM AUTHOR]