  We derive exact solutions of a linear form of the Grad-Shafranov (GS)
equation, including incompressible equilibrium flow, using ansatz-based
similarity reduction methods. The linearity of the equilibrium equation allows
linear combinations of solutions in order to obtain axisymmetric MHD equilibria
with closed and nested magnetic surfaces which are favorable for the effective
confinement of laboratory plasmas. In addition, employing the same reduction
methods we obtain analytical solutions for several non-linear forms of the GS
equation. In this context analytic force-free solutions in both linear and
nonlinear regimes are also derived.
