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In this paper we examine a problem connected with Cauchy's problem for linear elliptic equations.Let be a bounded region of , and let be its boundary. In we consider the elliptic equation (1)where is a regular elliptic expression with complex coefficients. Let , be a piece of the surface . The coefficients of the expression , the surface , and the boundary are assumed to be infinitely smooth. We are concerned with Cauchy's problem on with the initial conditions , , where designates the direction normal to . In this paper we prove that under our assumptions the set of Cauchy initial data for solutions of (1) in is dense in for any integer if Cauchy's problem is unique for the formal conjugate operator , as is the case, for example, when has no multiple complex characteristics.In addition, in this paper we give conditions under which the analogous assertion holds for certain elliptic systems.Bibliography: 4 items.