Abstract: An adaptation of the conditional symmetry method to the
case of kth order PDEs in many
independent variables is proposed. The necessary and sufficient conditions for
existence of conditional symmetry algebras for these PDEs are determined. In
order to facilitate solving these conditions we make use of singularity analysis
of the original PDE and submit this PDE to a specific Darboux transformation. A
new procedure for constructing conditionally invariant solutions is proposed,
based on a specific factorized form of multiple differential constraints. This
procedure establishes a link between conditional symmetries and Auto- Bäcklund
transformations. The developed approach is applied to the Tzitzéica equation
for which an Auto-Bäcklund transformation is reconstructed. It is defined by
projective matrix Riccati equations based on the algebra.
