Bi-Integrable Couplings Associated with so(3,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(3,{\mathbb R})$$\end{document}

By a class of zero curvature equations over a non-semisimple matrix loop algebra, we generate a new hierarchy of bi-integrable couplings for a soliton hierarchy associated with so(3, R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document}). The bi-Hamiltonian structures are found by the associated variational identity, which imply that all the presented coupling systems possess infinitely many commuting symmetries and conserved functionals and, thus, are Liouville integrable.