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 $lambdaequiv 2pisqrt{G} Sigma^{(g)}/B_z$ 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 $fequivbar{Phi}/Phi$ 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.
