Geodesic
Integrability conditions for systems of two equations of the form $u_t+A(u)u_{xx}+F(u,u_x)$. I
Teoretičeskaâ i matematičeskaâ fizika, Tome 62 (1985) no. 2, pp. 163-185 
A. V. Mikhailov; A. B. Shabat. Integrability conditions for systems of two equations of the form $u_t+A(u)u_{xx}+F(u,u_x)$. I. Teoretičeskaâ i matematičeskaâ fizika, Tome 62 (1985) no. 2, pp. 163-185. http://geodesic.mathdoc.fr/item/TMF_1985_62_2_a0/
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     title = {Integrability conditions for systems of two equations of the form $u_t+A(u)u_{xx}+F(u,u_x)${.~I}},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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