In this talk, we focus on the reduction of the 2-dimensional Toda hierarchy. Based on the Gauss-Borel decomposition of the moment matrix and appropriate time evolution, we construct the corresponding Lax representation. Symmetric moment matrices are considered that lead to the differential relations between symmetric tau functions and 2d-Toda’s tau function. Motivated by the Cauchy two-matrix model, we study the rank one shift condition in the symmetric case, from which the C-Toda lattice hierarchy is found with the special Lax matrices and intergrable structure.