The concept of quasi self-adjointness generalizes Definition \[saeq:def\] and is more convenient for dealing with systems (\[sa:eveq.int.1\]). This concept was formulated in [@ibr07d] as follows.

The system (\[sa:eveq.int.1\]) is *quasi self-adjoint* if the adjoint system (\[sa:eveq.int.2\]) becomes equivalent to the original system (\[sa:eveq.int.1\]) upon a substitution $$\label{qsa0.eq1}
 v = \varphi(u)$$ such that its derivative does not vanish in a certain domain of $u,$ $$\label{qsa0.eq2}
 \varphi'(u)  \not= 0, \quad {\rm where} \quad \varphi'(u) =
 \Big|\!\Big|\frac{\partial \varphi^\alpha(u)}{\partial u^\beta}\Big|\!\Big|.$$

\[qsa0:rem1\] The substitution (\[qsa0.eq1\]) defines a mapping $$v^\alpha = \varphi^\alpha(u), \quad \alpha = 1, \ldots, m,$$ from the $m$-dimensional space of variables $u = (u^1, \ldots,
u^m)$ into the $m$-dimensional space of variables $v = (v^1,\ldots, v^m).$ It is assumed that this mapping is continuously differentiable. The condition (\[qsa0.eq2\]) guarantees that it is invertible, and hence Eqs. (\[sa:eveq.int.2\]) and (\[sa:eveq.int.1\]) are equivalent. The equivalence means that the following equations hold with certain coefficients $\lambda^\beta_\alpha:$ $$\label{qsa.2}
 F^*_{\alpha} \big(x, u, \varphi, \ldots, u_{(s)}, \varphi_{(s)}\big) =
 \lambda^\beta_\alpha \, F_{\beta}\big(x, u,
 \ldots, u_{(s)}\big), \quad \alpha = 1, \ldots, m,$$ where $$\label{qsa.2notat}
   \varphi = \{\varphi^\alpha(u)\}, \quad \varphi_{(\sigma)}
  = \{D_{i_1} \cdots D_{i_\sigma}\big(\varphi^\alpha(u)\big)\}, \quad \sigma = 1, \ldots, s.$$ It can be shown that the matrix $\|\lambda^\beta_\alpha\|$ is invertible due to 