First, we prove that the BMO condition by John–Nirenberg leads in the natural way to the asymptotic homogeneity at the origin of regular homeomorphic solutions of the degenerate Beltrami equations. Then, on this basis we establish a series of criteria for the existence of regular homeomorphic solutions of the degenerate Beltrami equations in the whole complex plane with asymptotic homogeneity at infinity. These results can be applied to the fluid mechanics in strongly anisotropic and inhomogeneous media because the Beltrami equation is a complex form of the main equation of hydromechanics.