  We find the twisted extensions of the symmetry algebra of the 2D Euler
equation in the vorticity form and use them to construct new Lax representation
for this equation. Then we generalize this result by considering the
transformation Lie--Rinehart algebras generated by finite-dimensional
subalgebras of the symmetry algebra and derive a family of Lax representations
for the Euler equation. The family depends on functional parameters and
contains a non-removable spectral parameter.
