The Euler system on a space curve

We consider flows of an inviscid medium on a space curve in a constant gravitational field (the Euler system). We discuss symmetries and differential invariants of the Euler system, and give their classification based on symmetries group of the system. Using differential invariants for this system, we obtain its quotient. The solutions of the quotient equation that are constant along characteristic vector field provide some solutions of the Euler system.
Joint work with Valentin Lychagin.