<p>We derive analytical general solutions to the anti-plane elasticity problem of an isotropic elastic elliptical inhomogeneity perfectly bonded to an infinite isotropic elastic matrix when either the matrix or the elliptical inhomogeneity is subjected to an arbitrary singularity. The two analytic functions characterizing the elastic fields in the elliptical inhomogeneity and in the matrix can be written expediently once the complex potential for the same singularity in an infinite homogeneous plane is known. The present formulae are independent of the specific nature of the singularity. The solution method, which in principle can be classified as the method of images, differs from that based on standard Laurent series expansions used in previous studies. Several typical examples are presented to demonstrate our solutions. The present solution method is also adapted to solve the anti-plane shear problem of an arbitrary singularity interacting with an anisotropic elastic elliptical inhomogeneity.</p>