the condition (\[qsa0.eq2\]).

\[qsa:exaITT\] The quasi self-adjointness of nonlinear wave equations of the form $$u_{tt}-u_{xx} = f(t, x, u, u_t, u_x)$$ is investigated in [@ibr-tor-tra10]. The results of the paper [@ibr-tor-tra10] show that, e.g. the equation $$\label{qsa.exaITT:eq1}
 u_{tt}-u_{xx} + u_t^2-u_x^2 = 0$$ is quasi self-adjoint and that in this case the substitution (\[qsa0.eq1\]) has the form $$\label{qsa.exaITT:eq2}
 v = {\rm e}^u.$$ Indeed, the adjoint equation to Eq. (\[qsa.exaITT:eq1\]) is written $$\label{qsa.exaITT:eq1q}
 v_{tt}-v_{xx} - 2 v u_{tt} - 2 u_t v_t + 2 v u_{xx} + 2 u_x v_x = 0.$$ After the substitution (\[qsa.exaITT:eq2\]) the left-hand side of Eq. (\[qsa.exaITT:eq1q\]) takes the form (\[qsa.2\]): $$\label{qsa.exaITT:eq1s}
 v_{tt}-v_{xx} - 2 v u_{tt} - 2 u_t v_t + 2 v u_{xx} + 2 u_x v_x =
 - {\rm e}^u [u_{tt}-u_{xx} + u_t^2-u_x^2].$$ It is manifest from Eq. (\[qsa.exaITT:eq1s\]) that $v$ given by (\[qsa.exaITT:eq2\]) solves the adjoint equation (\[qsa.exaITT:eq1q\]) if one replaces $u$ by any solution of Eq. (\[qsa.exaITT:eq1\]).

In constructing conservation laws one can relax the condition (\[qsa0.eq2\]). Therefore I generalize the previous definition of quasi self-adjointness as follows.

\[qsa:def\] The system (\[sa:eveq.int.1\]) is said to be *quasi self-adjoint* if the adjoint equations (\[sa:eveq.int.2\]) are satisfied for all solutions $u$ of the original 