Two types of new Lie algebras and corresponding hierarchies of evolution equations

An extension of Lie algebra An−1 was proposed [Phys. Lett. A 310 (2003) 19, J. Math. Phys. 26 (1985) 1189]. Based on that extension, we presented two new Lie algebras. Meanwhile we obtained the corresponding hierarchies of evolution equations and it was shown that the corresponding hierarchies of evolution equations were integrable in Liouville sense. Finally we found integrable couplings of the second equations hierarchy based on a new loop algebra G∼ which was different from A2∼ in Refs. [Chaos, Solitons & Fractals 7 (1996) 1227, Phys. Lett. A 310 (2003) 19].