In this chapter, we replace the strong ellipticity condition of the previous chapter with a weaker ellipticity condition expressed only in terms of the symmetric matrices with trace zero. We call this condition the “relaxed ellipticity condition.” Our systems will now be formulated either in a bounded Lipschitz domain or in the exterior of a bounded Lipschitz open set (i.e., an exterior domain) in \({\mathbb R}^{n}\) , \(n \ge 3\) . Unlike in the previous chapter, in this chapter we also impose a symmetry condition on our viscosity coefficient tensor. We continue to use anisotropic, \(L^{\infty }\) viscosity coefficients tensors. We then prove the well-posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains in \({\mathbb R}^n\) with the data in \(L^2\) -based weighted Sobolev spaces. These results are used to define the Newtonian (or volume) potential operators and the layer potential operators in terms of solutions of the transmission problems and to obtain their properties. We also analyze the well-posedness of the exterior Dirichlet and Neumann problems for the anisotropic Stokes system and represent their solutions in terms of the Newtonian and layer potentials.