In 2006, Kostant and Wallach constructed an integrable system on the n x n complex matrices $M_{n}( C)$ using Gelfand-Zeitlin theory. This system can be viewed as a complexified version of the one studied by Guillemin and Sternberg on the n x n Hermitian matrices, which is related to the classical Gelfand-Zeitlin basis for irreducible representations of the unitary group via geometric quantization. In this talk, we discuss joint work with Sam Evens in which we develop a geometric description of the fibres of the moment map for the complexified Gelfand-Zeitlin system. Our approach uses the theory of orbits of a symmetric subgroup K of the group G of all invertible n x n complex matrices on the flag variety of $M_{n}( C)$ . These orbits play a central role in the geometric construction of Harish-Chandra modules for the pair $(M_{n} (C ), K)$ using the Beilinson-Bernstein correspondence. We indicate how our work provides the foundation for the geometric construction of a category of generalized Harish-Chandra modules studied by Drozd, Futorny, and Ovsienko.