Description of a helical motion of an incompressible nonviscous fluid

We consider a problem of describing the motion of a fluid filling at any specific instant $t\ge0$ a domain $D\subset R^3$ in terms of velocity $\mathbf v$ and pressure $p$. We assume that the pair of variables $(\mathbf v,p)$ satisfies a system of equations that includes Euler's equation and the incompressible fluid continuity equation. For the case of an axially symmetric cylindrical layer $D$, we find a general solution of this system of equations in the class of vector fields $\mathbf v$ whose lines for any $t\ge0$ coincide everywhere in $D$ with their vortex lines and lie on axially symmetric cylindrical surfaces nested in $D$. The general solution is characterized in a theorem. As an example, we specify a family of solutions expressed in terms of cylindrical functions, which, for $D=R^3$, includes a particular solution obtained for the first time by I. S. Gromeka in the case of steady-state helical cylindrical motions.