  Kuzmin-Oseledets formulations of compressible Euler equations case are
considered. Exact results and physical interpretations are given. One such
exact result for the compressible barotropic case is the potential helicity
Lagrange invariant. In recognition of the fundamental physical implications of
this Lagrange invariant clarified here, this invariant is shown to hold for
compressible non-barotropic cases as well upon using a stronger gauge
condition. Symmetry restoration taking place at the Lagrangian level in the
Kuzmin-Oseledets formulation is pointed out. The Kuzmin-Oseledets formulation
in the compressible barotropic case is shown to admit an exact solution that
physically describes a density wave on a steady irrotational flow with the
Kuzmin-Oseledets velocity q growing monotonically with it and hence specifying
some information about the fluid impulse that is needed to set up the flow in
question.
