The elliptic extension of GinSE considered in the previous chapter gives rise to a potential which is not spherically symmetric. In contrast, the extensions of GinSE to be considered in this chapter—specifically the induced GinSE relating to a polar decomposition involving a rectangular generalisation of GinSE and a random Haar symplectic unitary matrix, the quaternion spherical ensemble of matrices \(G_1^{-1}G_2\) with \(G_1,G_2\) both GinSE matrices, a sub-block of a Haar symplectic unitary matrix, and products of GinSE matrices—all have the common feature that the corresponding potential is spherically symmetric. This allows for a unified treatment in relation to the corresponding skew-orthogonal polynomials. Furthermore, in each case the pre-kernel can be shown to satisfy an inhomogeneous partial differential equation, from which scaled limits can be computed.