Symmetries and invariant solutions of the two-dimensional variable coefficient Burgers equation

We discuss symmetries and reductions of the two-dimensional Burgers equatio n with variable coefficient. We classify one-dimensional and two-dimensiona l subalgebras of the Burgers symmetry algebra which is infinite-dimensional into conjugacy classes under the adjoint action of the symmetry group. Inv ariance under one-dimensional subalgebras provides reductions to lower-dime nsional partial differential equations. Further reductions of these equatio ns to second order ordinary differential equations are obtained through inv ariance under two-dimensional subalgebras. The reduced ODEs are then analys ed and shown that they belong to the polynomial class of second-order equat ions which can be linearized only for particular values of parameters figur ing in the coefficient.