The exotic conformal Galilei algebra and nonlinear partial differential equations

The conformal Galilei algebra (cga) and the exotic conformal Galilei algebra (ecga) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the cga but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under cga. It is further shown that there are systems of nonlinear PDEs admitting ecga with the realisation obtained very recently in [D. Martelli, Y. Tachikawa, Comments on Galilei conformal field theories and their geometric realisation, preprint, arXiv:0903.5184v2 [hep-th], 2009]. Moreover, wide classes of nonlinear systems, invariant under two different 10-dimensional subalgebras of ecga are explicitly constructed and an example with possible physical interpretation is presented.