  We consider uniformly strongly elliptic systems of the second order with
bounded coefficients. First, sufficient conditions for the invariance of convex
bodies obtained for linear systems without zero order term in bounded domains
and quasilinear systems of special form in bounded and in a class of unbounded
domains. These conditions are formulated in algebraic form. They describe
relation between the geometry of the invariant convex body and the coefficients
of the system. Next, necessary conditions, which are also sufficient, for the
invariance of some convex bodies are found for elliptic homogeneous systems
with constant coefficients in a half-space. The necessary conditions are
derived by using a criterion on the invariance of convex bodies for normalized
matrix-valued integral transforms also obtained in the paper. In contrast with
the previous studies of invariant sets for elliptic systems no a priori
restrictions on the coefficient matrices are imposed.
