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A numerically stable algorithm is derived to compute orthonormal bases for any deflating subspace of a regular pencil . The method is based on an update of the -algorithm, in order to obtain any desired ordering of eigenvalues in the quasitriangular forms constructed by this algorithm. As applications we discuss a new approach to solve Riccati equations arising in linear system theory. The computation of deflating subspaces with specified spectrum is shown to be of crucial importance here.