Integrable systems in spaces of arbitrary dimension

TL;DR

This paper introduces a framework for multidimensional integrable systems on 2n-dimensional manifolds, connecting them with graded semisimple algebras and providing explicit general solutions.

Contribution

It constructs new multidimensional integrable systems linked to arbitrary graded semisimple algebras and derives their explicit general solutions.

Findings

Introduction of a 2n-dimensional manifold with commuting differential operators

Construction of integrable systems associated with graded semisimple algebras

Explicit general solutions for these systems provided

Abstract

The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general solution of them is presented in explicit form.