Symmetries of scalar fields. I

A definition of the generating operator of a system of nonlinear differential equations is proposed, and the connection between such operators and Lie–Bäcklund algebras is established. For classical nonlinear scalar fields in $n$-dimensional ($n>2$) space-time interacting through a potential the Lie–Bäcklund algebra is investigated, and it is concluded that there are no differential generating operators. It is shown that in nonlinear theory in $n$-dimensional ($n>2$) space-time the number of independent local conservation laws is always finite.