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