A solution class of the Euler equation in a torus with solenoidal velocity field

A system of equations with respect to a pair $(\mathbf V,p)$ of a scalar field and a vector field in a torus $D$ is considered. The system consists of the Euler equation with a given vector field $\mathbf f$ and the solenoidality equation for the field $\mathbf V$. We seek for solutions $(\mathbf V,p)$ of this system for which lines of the vector field $\mathbf V$ inside $D$ coincide with meridians of tori embedded in $D$ with the same circular axis. Conditions on the vector field $\mathbf f$ under which the problem is solvable are established, and the whole class of such solutions is described.