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We study an iso-spectral deformation of general matrix which is a naturalgeneralization of the Toda lattice equation. We prove the integrability of thedeformation, and give an explicit formula for the solution to the initial valueproblem. The formula is obtained by generalizing the orthogonalizationprocedure of Szeg\"{o}. Based on the root spaces for simple Lie algebras, weconsider several reductions of the hierarchy. These include not only theintegrable systems studied by Bogoyavlensky and Kostant, but also theirgeneralizations which were not known to be integrable before. The behaviors ofthe solutions are also studied. Generically, there are two types of solutions,having either sorting property or blowing up to infinity in finite time.Comment: 25 pages, AMSLaTe

Iso-spectral deformations of general matrix and their reductions on Lie algebras