  In toroidally confined plasmas, the Grad-Shafranov equation, in general a
non-linear PDE, describes the hydromagnetic equilibrium of the system. This
equation becomes linear when the kinetic pressure is proportional to the
poloidal magnetic flux and the squared poloidal current is a quadratic function
of it. In this work, the eigenvalue of the associated homogeneous equation is
related with the safety factor on the magnetic axis, the plasma beta and the
Shafranov shift, then, the adjustable parameters of the particular solution are
bounded through physical constrains. The poloidal magnetic flux becomes a
linear superposition of independent solutions and its parameters are adjusted
with a non-linear fitting algorithm. This method is used to find hydromagnetic
equilibria with normal and reversed magnetic shear and defined values of the
elongation, triangularity, aspect-ratio, and X-point(s). The resultant toroidal
and poloidal beta, the safety factor at the $95\%$ flux surface and the plasma
current are in agreement with usual experimental values for high beta
discharges and the model can be used locally to describe reversed magnetic
shear equilibria.
