Study on systems of partial differential equations applying the abstract algebra

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(1) We have shown that, for the Cauchy=Kowalevskaya theorem on a system, it is necessary that the system is similar to Kowalevskian accepting poles on symbols. On the sufficiency, we have succeeded to prove it on some models, but we need refine on the normal form in order to treat the general case. (2) On the systems with coefficients depending only on the time variable, it is necessary and sufficient for the strong hyperbolicity that the system is uniformly diagonalizable as a hyperbolic Fuchsian. (3) The study under the new definition of the p-parabolic system is not satisfactory.