Toda lattice representation for random matrix model with logarithmic confinement

We construct a replica field theory for a random matrix model with logarithmic confinement [K.A.Muttalib et.al., Phys. Rev. Lett. 71, 471 (1993)]. The corresponding replica partition function is calculated exactly for any size of matrix $N$. We make a color-flavor transformation of the original model and find corresponding Toda lattice equations for the replica partition function in both formulations. The replica partition function in the flavor space is defined by generalized Itzikson-Zuber (IZ) integral over homogeneous factor space of pseudo-unitary supergroups $SU(n\mid M,M)/SU(n\mid M-N,M)$ (Stiefel manifold) with $M\to\infty$, which is evaluated and represented in a compact form.