The equilibrium of an axisymmetric toroidal plasma in an external field (a so-called hybrid system) with free boundary and arbitrary cross-section is solved analytically in a curvilinear co-ordinate system by means of an expansion in the inverse aspect ratio. The results are expressed as general relation s between Fourier coefficients of flux functions. An expression for the vertical field maintaining non-circular plasma equilibrium and th e condition for beta-free confinement, under which the strength of the vertical field does not depend on plasma pressure, are derived. Some typical configurations are calculated numerically.