  We find a remarkable subalgebra of higher symmetries of the elliptic
Euler-Darboux equation. To this aim we map such equation into its hyperbolic
analogue already studied by Shemarulin. Taking into consideration how
symmetries and recursion operators transform by this complex contact
transformation, we explicitly give the structure of this Lie algebra and prove
that it is finitely generated. Furthermore, higher symmetries depending on jets
up to second order are explicitly computed.
