The noncommutative KdV equation and its para-Kähler structure

We prove that the noncommutative (n × n)-matrix KdV equation is exactly a reduction of the geometric KdV flows from ℝ to the symmetric para-Grassmannian manifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde G_{2n,n} $\end{document} = SL(2n, ℝ)/SL(n, ℝ) × SL(n, ℝ) and it can also be realized geometrically as a motion of Sym-Pohlmeyer curves in the symmetric Lie algebra sl(2n, ℝ) with initial data being suitably restricted. This gives a para-geometric characterization of the noncommutative matrix KdV equation.