1. Introduction  
2. Background  
3. Nagy-Foias Theory in \(H^{2}(\mathbb{D}^{2})\)  
4. Commutators  
5. Two-variable Jordan Block  
6. Fredholmness of the Pairs \((R_{1}, R_{2})\) and \((S_{1}, S_{2})\)  
7. Essential Normality of Quotient Module  
8. Two Single Companion Operators  
9. Congruent Submodules and Their Invariants  
10. Concluding Remarks