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 $-1/2$.

我们对二维退化 Burgers 方程进行扩展对称分析。用代数方法找到了它的完全点对称群，证明了它所有的广义对称性等价于它的李对称性。我们还证明了该方程的守恒定律空间是无限维的，并且与(1+1)维后向线性热方程的解空间自然同构。以最优方式综合研究了二维退化Burgers方程的李约简，构造了新的李不变解。我们另外考虑也满足类似的非退化 Burgers 方程的解。总共，我们构造了四类二维退化伯格斯方程的解，这些方程用 (1+1) 维线性热方程的任意（非零）解表示。还讨论了各种隐藏的对称性和隐藏的守恒定律（局部和潜在的）。作为副产品，我们详尽地描述了输运方程（也称为无粘伯格斯方程）的广义对称性、共对称性和守恒定律，并构建了非线性扩散和具有阶次幂非线性的扩散-对流方程的新不变解$-1/2$.