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Recently Strahov has extended Tracy and Widom's Fredholm theory of the hard edge of single random matrices with unitary symmetry ($ M=1 $) to products of matrices ($ M>1 $). In this earlier theory it was discovered that certain solutions to Painlev\'e's third transcendent were central in determining the distribution of the lowest eigenvalue (always non-negative) of the random matrix ensemble. In the recent work some kind of generalisation of this system plays the same role and we explore some of the properties of this integrable system in particular the first extension $ M=2 $. This is joint work with Peter Forrester.