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

Necessary conditions for the existence of nontrivial conservation laws for systems of nonlinear equations of the form $u_t=a(u,v)u_{xx}+b(u,v)v_{xx}+f(u,v,u_x,v_x),$ $-v_t=c(u,v)u_{xx}+d(u,v)v_{xx}+g(u,v,u_x,v_x)$. are found. They take the form of densities of local conservation laws constructed in a definite manner from the coefficients of the system. The conditions can be readily verified in each specific case. A module of simple invertible substitutions that makes it possible to reduce the system to a canonical form when some integrability conditions are satisfied is discussed.