Structure of magnetic fields in non-convective stars

(Abridged) We develop a theoretical framework to construct axisymmetric magnetic equilibria in stars, consisting of both poloidal and toroidal magnetic field components. In a stationary axisymmetric configuration, the poloidal current is a function of the poloidal magnetic flux only, and thus should vanish on field lines extending outside of the star. Non-zero poloidal current is limited to a set of toroid-shape flux surfaces fully enclosed inside the star. If we demand that there are no current sheets, then on the separatrix delineating the regions of zero and finite toroidal magnetic field both the poloidal flux function and its derivative should match. Thus, for a given magnetic field in the bulk of the star, the elliptical Grad-Shafranov equation that describes magnetic field structure inside the toroid is an ill-posed problem, with both Dirichlet and Newman boundary conditions and {\it a priori} unknown distribution of toroidal and poloidal electric currents. We discuss a procedure which allows to solve this ill-posed problem by adjusting the unknown current functions. We find a poloidal current-carrying solution that leaves the shape of the flux function and, correspondingly, the toroidal component of the electric current the same as in the case of no poloidal current. The equilibria discussed in this paper may have arbitrary large toroidal magnetic field, and may include a set of stable equilibria. The method developed here can also be applied to magnetic structure of differentially rotating stars, as well as to calculate velocity field in incompressible isolated fluid vortex with a swirl.