  (Abridged) We develop an analytical spectral method to solve the equations of
equilibrium for a self-gravitating, magnetized fluid body, under the only
hypotheses that (a) the equation of state is isothermal, (b) the configuration
is scale-free, and (c) the body is electrically neutral. All physical variables
are represented as series of scalar and vector spherical harmonics of degree l
and order m, and the equilibrium equations are reduced to a set of coupled
quadratic algebraic equations for the expansion coefficients of the density and
the magnetic vector potential. The method is general, and allows to recover
previously known hydrostatic and magnetostatic solutions possessing axial
symmetry. A linear perturbation analysis of the equations in spectral form show
that these basic axisymmetric states, considered as a continuos sequence with
the relative amount of magnetic support as control parameter, have in general
no neighboring nonaxisymmetric equilibria. This result lends credence to a
conjecture originally made by H. Grad and extends early results obtained by E.
Parker to the case of self-gravitating magnetized bodies. The only allowed
bifurcations of this sequence of axisymmetric equilibria are represented by
distortions with dipole-like angular dependence (l=1) that can be continued
into the nonlinear regime. These new configurations are either (i) azimuthally
asymmetric (m=+-1) or (ii) azimuthally symmetric but without reflection
symmetry with respect to the equatorial plane (m=0). To the extent that
interstellar clouds can be represented as isolated magnetostatic equilibria,
the results of this study suggest that the observed triaxial shapes of
molecular cloud cores can be interpreted in terms of weakly damped Alfven
oscillations about an equilibrium state.
