  We obtain well-posedness results in $L_p$-based weighted Sobolev spaces for a
transmission problem for anisotropic Stokes and Navier-Stokes systems with
$L_{\infty}$ strongly elliptic coefficient tensor, in complementary Lipschitz
domains of ${\mathbb R}^n$, $n\ge 3$. The strong ellipticity allows to explore
the associated pseudostress setting. First, we use a variational approach that
reduces two linear transmission problems for the anisotropic Stokes system to
equivalent mixed variational formulations with data in $L_p$-based weighted
Sobolev and Besov spaces. We show that such a mixed variational formulation is
well-posed in the space ${\mathcal H}^1_{p}({\mathbb R}^n)^n\times L_p({\mathbb
R}^n)$, {$n\geq 3$}, for any $p$ in an open interval containing $2$. These
results are used to define the Newtonian and layer potential operators for the
considered anisotropic Stokes system. Various mapping properties of these
operators are also obtained. The potentials are employed to show the
well-posedness of some linear transmission problems, which then is combined
with a fixed point theorem in order to show the well-posedness of the nonlinear
transmission problem for the anisotropic Stokes and Navier-Stokes systems in
$L_p$-based weighted Sobolev spaces, whenever the given data are small enough.
