Symmetry algebras of linear differential equations

The local symmetries of linear differential equations are investigated by means of proven theorems on the structure of the algebra of local symmetries of translationatly and dilatationally invariant differential equations. For a nonparabolic second-order equation, the absence of nontrivial nonlinear local symmetries is proved. This means that the local symmetries reduce to the Lie algebra of linear differential symmetry operators. For the Laplace–Beltrami equation, all local symmetries reduce to the enveloping algebra of the algebra of the conformal group.