Abstract. Let Ω 0 be an ellipsoidal inclusion in the Euclidean space R n . It is checked that if a solution of the homogeneous transmission problem for a formally selfadjoint elliptic system of second order differential equations with piecewise smooth coefficients grows linearly at infinity, then this solution is a linear vector-valued function in the interior of Ω 0 . This fact generalizes the classical Eshelby theorem in elasticity theory and makes it possible to indicate simple and explicit formulas for the polarization matrix of the inclusion in the composite space, as well as to solve a problem about optimal patching of an elliptical hole. §1. Setting of the transmission problem Let Ω 0 be a nonempty domain in the Euclidean space R n , n ≥ 2, with a smooth (of class C ∞ ) boundary, and let Ω 1 = R n \ Ω 0 be its exterior. Suppose it has compact closure:We place the origin of a Cartesian coordinate system x = (x 1 , . . . , x n ) in the interior of Ω 0 . Let D(∇ x ) be an (N ×k)-matrix of first order differential operators with constant and, in general, complex coefficients, and let A i , i = 0, 1, be numerical Hermitian positive definite matrices of size N × N . We introduce the formally selfadjoint (k × k)-matrices of second order differential operatorsand consider the following system of differential equations with transmission conditions on the interface Γ = ∂Ω 0 = ∂Ω 1 :