We continue the analysis of the anisotropic Stokes system with \(L^{\infty }\) viscosity coefficient tensor satisfying the relaxed ellipticity and symmetry conditions of Chap. 6 (i.e., in terms of only symmetric matrices with vanishing trace) and obtain existence and uniqueness results for Dirichlet problems and Dirichlet-transmission problems in bounded Lipschitz domains, as well as in exterior Lipschitz domains of \({\mathbb R}^n\) (with \(n \geq 3\) in the case of exterior domains), by means of variational arguments. Then, we consider a Dirichlet problem and a Dirichlet-transmission problem in bounded Lipschitz domains of \({\mathbb R}^3\) for the Navier-Stokes system and prove the corresponding existence result for large data. In addition, we prove that the solution of the Dirichlet-transmission problem is unique for small data. Finally, we obtain existence results for similar boundary value problems for the Navier-Stokes system in an exterior domain and for a transmission problem for the same system in \({\mathbb R}^3\) .