  Let A be a domain of the boundary of a strictly pseudoconvex domain \Omega of
C^n and M a smooth, closed, maximally complex submanifold of A. We find a
subdomain \widetilde A of \Omega, depending only on \Omega and A, and a complex
subvariety W of \widetilde A such that bW\cap A = M. Moreover, a generalization
to analytic sets of depth at least 4 is given.
