  We study global non-axisymmetric stationary perturbations of aligned and
unaligned logarithmic spiral configurations in an axisymmetric composite
differentially rotating disc system of scale-free stellar and isopedically
magnetized gas discs coupled by gravity. The gas disc is threaded across by a
vertical magnetic field $B_z$ with a constant dimensionless isopedic ratio
$\lambda\equiv 2\pi\sqrt{G} \Sigma^{(g)}/B_z$ of surface gas mass density
$\Sigma^{(g)}$ to $B_z$ with $G$ being the gravitational constant. Our
exploration focuses on the relation between $\lambda$ and the dark matter
amount represented by a ratio $f\equiv\bar{\Phi}/\Phi$ in order to sustain
stationary perturbation configurations, where $\bar{\Phi}$ is the gravitational
potential of a presumed axisymmetric halo of dark matter and $\Phi$ is the
gravitational potential of the composite disc matter. High and low $\lambda$
values correspond to relatively weak and strong magnetic fields given the same
gas surface mass density, respectively. The main goal of our model analysis is
to reveal the relation between isopedic magnetic fields and dark matter halo in
spiral galaxies with globally stationary perturbation configurations. Our
results show that, fairly strong yet realistic magnetic fields require a
considerably larger amount of dark matter in aligned and unaligned cases than
weak or moderate magnetic field strengths. We discuss astrophysical and
cosmological implications of our findings. For examples, patterns and pattern
speeds of galaxies may change during the course of galactic evolution.
Multiple-armed galaxies may be more numerous in the early Universe. Flocculent
galaxies may represent the transitional phase of pattern variations in
galaxies.

Cosmology and Nongalactic Astrophysics