  The first aim of this paper is to develop a layer potential theory in
$L_2$-based weighted Sobolev spaces on Lipschitz bounded and exterior domains
of ${\mathbb R}^n$, $n\geq 3$, for the anisotropic Stokes system with
$L_{\infty }$ viscosity coefficient tensor satisfying an ellipticity condition
for symmetric matrices. To do this, we explore equivalent mixed variational
formulations and prove the well-posedness of some transmission problems for the
anisotropic Stokes system in Lipschitz domains of ${\mathbb R}^n$, with the
given data in $L_2$-based weighted Sobolev spaces. These results are used to
define the Newtonian and layer potentials and to obtain their properties. Then
we analyze well-posedness of the exterior Dirichlet, Neumann and mixed problems
for the Stokes system with $L_{\infty }$ symmetrically elliptic coefficient
tensor. Solutions of some of these problems are also represented in terms of
the anisotropic Stokes Newtonian and layer potentials. Finally, we prove the
existence of a weak solution for a transmission problem in complementary
Lipschitz domains in ${\mathbb R}^3$ for the anisotropic Navier-Stokes system
with general data in $L_2$-based weighted Sobolev spaces. The analysis relies
on an existence result for a Dirichlet problem for the anisotropic
Navier-Stokes system in a family of bounded domains, and on the Leray-Schauder
fixed point theorem.
