Abstract A study is made of inverse problems for n × n systems of the form L( λ) = Mλ 2 + Dλ + K. This paper concerns the determination of systems in an equivalence class defined by a fixed 2 n × 2 n admissible Jordan matrix, i.e. a class of isospectral systems. Constructive methods are obtained for complex or real systems with no symmetry constraints. It is also shown how isospectral families of complex hermitian matrices can be formed. The case of real symmetric matrices is more difficult. Some partial solutions are obtained but, in this case, the theory remains incomplete. Examples are given.