Group classification and compatibility analysis of equations describing helical flows of an ideal incompressible fluid

We consider compatibility conditions for equations describing nonuniform helical flows of an inviscid incompressible fluid. The system under study includes the Euler equations supplemented by differential constraints defining the helical flows. In the general case of an arbitrary function establishing relationships between the velocity and the vorticity vector, this system is not involutive. Since reducing this overdetermined system to involutive form generally leads to cumbersome calculations, the study focuses on functions identified as a result of preliminary group classification. This classification leads to several nonequivalent cases. We carry out a complete study of compatibility of the cases where the factor algebra modulo the kernel of the Lie algebra corresponding to the equivalence group has dimension greater than $$2$$ .