  We consider the Euler system describing a one-dimensional inviscid flows in
space along curves of a certain class. Using differential invariants for the
Euler system, we obtain its quotient equation. The solutions of the quotient
equation that are constant along characteristic vector field provide some
solutions of the Euler system. We discuss solving the quotient using asymptotic
expansions of unknown functions and virial expansion of thermodynamic state
equations. Thus the quotient is reduced to a series of ODE systems.
