  We investigate the equations of anisotropic incompressible viscous fluids in
$\R^3$, rotating around an inhomogeneous vector $B(t, x_1, x_2)$. We prove the
global existence of strong solutions in suitable anisotropic Sobolev spaces for
small initial data, as well as uniformlocal existence result with respect to
the Rossby number in the same functional spaces under the additional assumption
that $B=B(t,x_1)$ or $B=B(t,x_2)$. We also obtain the propagation of the
isotropic Sobolev regularity using a new refined product law.
