where A(x, Dx) is a second-order elliptic differential operator in O, and v is a smooth non-vanishing real vector field on F. When v is nowhere tangent to F, the problem (0.1) is, so-called, of coercive type and satisfactory results are obtained (e.g., see [7]). Egorov and Kondrat'ev in [2] have considered (0.1) when v is tangent to F on its submanifold F0, and have classified the problem into three cases in the following way. First class: v leaves Q through F0; Second class: v enters Q through F0; Third class: v neither leaves nor enters Q through F0, (for details, see [2] or § 1 of our paper). In the first class the problem (0.1) has an infinite-dimensional kernel. Therefore, adding the Dirichlet condition w|Fo to (0.1), they have shown that the problem (A(x, Dx)u=f in Q, du W = 9 on F, u = ho on Fn