the variables $x, u, u_{(1)}, u_{(2)}, \ldots$ is called a *differential function*. The set of all differential functions is denoted by ${\cal A}.$ For more details see [@ibr99], Chapter 8.

Linear self-adjointness {#lsa}
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Recall that the adjoint operator $F^*$ to a linear operator $F$ in a Hilbert space $H$ with a scalar product $(u, v)$ is defined by $$\label{ad.H}
 (Fu, v) = (u, F^*v), \quad u, v \in H.$$

Let us consider, for the sake of simplicity, the case of one dependent variable $u$ and denote by $H$ the Hilbert space of real valued functions $u(x)$ such that $u^2(x)$ is integrable. The scalar product is given by $$(u, v) = \int_{\footnotesize \R^n} u(x) v(x) dx.$$ Let $F$ be a linear differential operator in $H.$ Its action on the dependent variable $u$ is denoted by $F[u].$ The definition (\[ad.H\]) of the adjoint operator $F^*$ to $F,$ $$(F[u], v) = (u, F^*[v]),$$ can be written, using the divergence theorem, in the simple form $$\label{clad}
 v F[u]-uF^*[v] = D_i(p^i),$$ where $v$ is a new dependent variable, and $p^i$ are any functions of $x,\,u,\,v,\, u_{(1)},\,v_{(1)},\ldots .$

It is manifest from Eq. (\[clad\]) that the operators $F$ and $F^*$ are mutually adjoint, $$\label{adad}
 \left(F^*\right)^* 