  We carry out the extended symmetry analysis of a two-dimensional degenerate
Burgers equation. Its complete point-symmetry group is found using the
algebraic method, and all its generalized symmetries are proved equivalent to
its Lie symmetries. We also prove that the space of conservation laws of this
equation is infinite-dimensional and is naturally isomorphic to the solution
space of the (1+1)-dimensional backward linear heat equation. Lie reductions of
the two-dimensional degenerate Burgers equation are comprehensively studied in
the optimal way and new Lie invariant solutions are constructed. We
additionally consider solutions that also satisfy an analogous nondegenerate
Burgers equation. In total, we construct four families of solutions of
two-dimensional degenerate Burgers equation that are expressed in terms of
arbitrary (nonzero) solutions of the (1+1)-dimensional linear heat equation.
Various kinds of hidden symmetries and hidden conservation laws (local and
potential ones) are discussed as well. As a by-product, we exhaustively
describe generalized symmetries, cosymmetries and conservation laws of the
transport equation, also called the inviscid Burgers equation, and construct
new invariant solutions of the nonlinear diffusion and diffusion-convection
equations with power nonlinearities of degree -1/2.
