In this dissertation, I give a detailed description of an ansatz of generalizing the construction of recursion relations for the correlation functions of the sl_2-invariant fundamental exchange model in the thermodynamic limit by Boos, Jimbo, Miwa, Smirnov, Takeyama in 2004 for higher rank. Due to the structure of the correlators as functions of their inhomogeneity parameters, a recursion formula for the reduced density matrix was proven. In the case of sl_3, I use the explicit results of Klümper and Ribeiro and Boos, Hutsalyuk and Nirov for the reduced density matrix of up to operator length three to verify whether it is possible to relate the residues of the density matrix of length n to the density matrix of length smaller than n as in sl_2. This is unclear, since the reduced quantum Knizhnik--Zamolodchikov equation splits into two parts for higher rank. In fact, I show two relations, one of which is analogous to the sl_2 case and one which is new. This allows me to construct an analogue of the operator X_k, which I call 'Snail Operator'. In the sl_2 case this operator has many nice properties including in particular the fact that only one irreducible representation of the Yangian Y(sl_2), the Kirillov--Reshetikhin module W_k, contributed the residue at lambda_i - lambda_j = - (k+1). Here, I give an overview of the mathematical background, T systems and show a new application of the extended T-systems introduced by Mukhin and Young in 2012 regarding the Snail Operator.