We consider an anisotropic body bounded by a cylindrical surface, which is infinitely long in the axial direction. Suppose the body is loaded in such a way that the field variables do not vary along the generators. An exact correspondence is established between the plane piezoelectric equations and those of generalized plane strain in elasticity. In particular, we show that by setting a linkage between the two sets of material constants, any problem of a plane deformation in piezoelectricity may be solved as a generalized plane strain in elasticity and vice versa. The assertation is true for rectilinearly, as well as for cylindrically, anisotropic solids. The equivalence is found for the most general anisotropic case, which links the fields between a monoclinic piezoelectric body of class m and a fully anisotropic (triclinic) elastic solid. A few degenerate systems are also identified. In addition, these correspondences can be extended to inhomogeneous media. Applied to composite materials or polycrystalline aggregates, they imply that results for effective elastic tensors immediately give the formulae for the effective electroelastic tensors (and vice versa). We also demonstrate that the connection can be used to extend the scope of the invariant stress theorem proved by Cherkaev, Lurie and Milton. In illustration, we present solutions for the plane problem of an elliptical inhomogeneity in an unbounded piezoelectric medium subjected to a uniform loading at infinity. Lekhnitskii's complex potential approach, together with the conformal mapping technique, are employed. General solutions of the fields inside the inhomogeneity and the matrix are obtained. The results are analytically proven to be identical with the existing solutions of the corresponding purely elastic boundary value problem.