  We study the stationary Stokes system with variable coefficients in the whole
space, a half space, and on bounded Lipschitz domains. In the whole and half
spaces, we obtain a priori $\dot W^1_q$-estimates for any $q\in [2,\infty)$
when the coefficients are merely measurable functions in one fixed direction.
For the system on bounded Lipschitz domains with a small Lipschitz constant, we
obtain a $W^1_q$-estimate and prove the solvability for any $q\in (1,\infty)$
when the coefficients are merely measurable functions in one direction and have
locally small mean oscillations in the orthogonal directions in each small
ball, where the direction is allowed to depend on the ball.
