  This paper studies the problem of finding a three-dimensional solenoidal
vector field such that both the vector field and its curl are tangential to a
given family of toroidal surfaces. We show that this question can be translated
into the problem of determining a periodic solution with periodic derivatives
of a two-dimensional linear elliptic second-order partial differential equation
on each toroidal surface, and prove the existence of smooth solutions. Examples
of smooth solutions foliated by toroidal surfaces that are not invariant under
continuous Euclidean isometries are also constructed explicitly, and they are
identified as equilibria of anisotropic magnetohydrodynamics. The problem
examined here represents a weaker version of a fundamental mathematical problem
that arises in the context of magnetohydrodynamics and fluid mechanics
concerning the existence of regular equilibrium magnetic fields and steady
Euler flows in bounded domains without continuous Euclidean isometries. The
existence of such configurations represents a key theoretical issue for the
design of the confining magnetic field in nuclear fusion reactors known as
stellarators.
