Integrability conditions for systems of two equations of the form $u_t=A(u)u_{xx}+F(u,u_x)$. II

The conditions (integrability conditions) that the right-hand side of any system of equations of the form indicated in the title must satisfy if the system is to have a rich set of conservation laws are found. The iutegrability conditions form an overdetermined system of nonlinear partial differential equations. This overdetermined system of equations can be integrated by quadrature and the form of the right-hand side completely determined.