Conditional Lie–Bäcklund symmetries and sign-invariants to second-order evolution equations

The present paper discusses a class of second-order evolution equations which generalize reaction-diffusion–convection equations. It is shown that some equations admit of second-order conditional Lie–Bäcklund symmetries and first-order Hamilton–Jacobi sign-invariants, which preserve both signs, ⩾ 0 and ⩽ 0 , on the solution manifold. Some examples are given to present the corresponding symmetry reductions, which reduce the considered equations to two-dimensional dynamical systems.