Summary: We consider the case of an anisotropic elastic elliptical inhomogeneity embedded in an infinite anisotropic elastic matrix subjected to a non-uniform remote loading described by remote stresses and strains which are linear functions of the two in-plane coordinates. The internal stresses and strains within the elliptical inhomogeneity are found to be linear functions of the two in-plane coordinates. In addition, we obtain explicit real-form solutions describing the elastic field inside the inhomogeneity as well as hoop stress vectors and hoop stresses on the matrix side and on the inhomogeneity side. We also obtain the corresponding solutions for the two limiting cases in which the elliptical inhomogeneity takes the form of a hole or a rigid inhomogeneity. The solution method presented here can be extended to accommodate the more general scenario in which the remote applied stresses and strains are arbitrary-order polynomials of the two in-plane coordinates.