Reaction–Diffusion on the Sphere with a Nonlinear Source Term: Symmetry Analysis, Group Classification, and Similarity Solutions

We consider the nonlinear reaction–diffusion equation on the unit sphere (Formula presented.), (Formula presented.), and carry out a complete Lie point symmetry analysis. Solving the associated determining system yields a rigidity theorem: for every genuinely nonlinear (Formula presented.), the admitted symmetry algebra is (Formula presented.), generated by the rotational Killing fields and time translation. We further show through a group classification that the source families that enlarge symmetries in Euclidean space do not produce any additional point symmetries on (Formula presented.). From an optimal system of subalgebras, we derive curvature-adapted reductions in which the Laplace–Beltrami operator becomes a Legendre-type operator in intrinsic invariants. For the specific nonlinear source (Formula presented.), specific reduced ODEs admit a hidden one-parameter symmetry, yielding a first integral and explicit steady states on (Formula presented.).