On the structure of $(2+1)$--dimensional commutative and noncommutative integrable equations

We develop the symbolic representation method to derive the hierarchies of $(2+1)$-dimensional integrable equations from the scalar Lax operators and to study their properties globally. The method applies to both commutative and noncommutative cases in the sense that the dependent variable takes its values in $\mathbb{C}$ or a noncommutative associative algebra. We prove that these hierarchies are indeed quasi-local in the commutative case as conjectured by Mikhailov and Yamilov in 1998. We propose a ring extension based on the symbolic representation. As examples, we give noncommutative versions of KP, mKP and Boussinesq equations.