The object of the investigation is a two-dimensional autonomous differential system whose right-hand sides are homogeneous polynomials of the fourth degree. The subject of the study is local phase portraits of this type of systems. The Frommer method used in the study is briefly described as one of the effective tools for splitting multiple equilibrium points, and a brief review of the known results for nonlinear systems is provided. The goal of the study is to obtain not only the local phase portraits of the systems under investigation but also the coefficient conditions under which such phase portraits exist. The relevance of the work is justified by the incompleteness of obtaining this type of coefficient criteria due to the difficulties in solving an algebraic equation of the fifth degree, which arises when finding possible finite measures of curvature of the $O^+$-curves of the system under study. In the main part of the paper, local phase portraits of the system under study are obtained for some cases and coefficient criteria for their implementation are indicated. The results of the research are summarized in the conclusion. The obtained results can be used in further research of two-dimensional autonomous systems.

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