The present paper develops a method for construction of exact solutions of biharmonic boundary-value problems of elasticity theory and to its application to a broad class of boundary-value problems with self-adjoint operators of order 2n in a rectangular region. The properties of the eigenfunctions are established and the problem of determining the coefficients in n-fold expansions of a special type is solved in closed form – in the form of generalized Fourier relations – using the resulting biorthogonality relations. A theorem of sufficient conditions for convergence of these expansions to a specified system of functions is proven. A constructive algorithm is presented for determination of the coefficients of n-fold expansions of general form of n different real functions in eigenelement series with one system of explicitly determined complex constants. A proof of n-fold completeness of the eigenelement system is given for the corresponding generalized eigenvalue problem.