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By assuming well defined magnetic surfaces we examine the characteristics of special three dimensional equilibria with incompressible flows such that any two out of the vectors of velocity, vorticity, magnetic field and current density are parallel one another. In the majority of the equilibria considered, it turns out that the only isodynamic equilibrium configuration, defined so that the magnetic field modulus be uniform on magnetic surfaces, is the axisymmetric Palumbo one. Also, except for equilibria with the velocity parallel to the current density in the other five of the above mentioned special axisymmetric (non-isodynamic) equilibria the velocity should be parallel to the magnetic field. In addition, isodynamicity is removed by the non parallel component of the flow associated with the electric field in axisymmetric equilibria with either complex lamellar generalised velocity, ρ V where ρ is the density, or complex lamellar magnetic field.

A Solovev-like equilibrium with arbitrary incompressible plasma flow nonparallel to the magnetic field is identified. The equilibrium configuration having a corner on the axis of symmetry resembles the nonlinear Palumbo one. The nonparallel flow associated with the quantity λ ∝ d / d u [ ρ ( d Φ / d u ) 2 ] (where ρ is the density, Φ the electrostatic potential and u labels the magnetic surfaces) affects the plasma shape, the Shafranov shift and the safety factor. For λ 0 the equilibrium has an x -point additional to the magnetic axis and increase of | λ | results in a variety of configurations with changing magnetic topology.

A sufficient condition for the linear stability of nonautonomous dissipative mechanical systems with circulatory forces is derived. It is applied to autonomous systems transformed to nonautonomous ones by a time-dependent orthogonal transformation. This allows to obtain sufficient stability conditions in case the perturbed potential energy can have negative values, which was not possible up to now. This is, in particular, of importance for plasma equilibria with sheared flows. The nearness of the sufficient condition to necessity is discussed for a particular example.

From the equilibrium solution of a toroidal plasma with toroidal velocity it is possible to define a relaxed energy state. This state corresponds to the minimum total energy at a given current and under certain assumptions retains the current and pressure profiles of the static case. For large toroidal radius it produces unique current and pressure profiles which however are completely different to those of the static case.

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., 5 pages, 3 figures