Journal of Mathematical Physics, Volume 61, Issue 9, September 2020. A correspondence between the orbits of a system of two complex, homogeneous, polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed. This correspondence is general in the sense that it applies to an open set of systems of ordinary differential equations of the specified kind. This allows us to transfer results well known from the theory of polygonal billiards, such as ergodicity, the existence of periodic orbits, the absence of exponential divergence, the existence of additional conservation laws, and the presence of discontinuities in the dynamics, to the corresponding systems of ordinary differential equations. It also shows that the considerable intricacy known to exist for polygonal billiards also attends these apparently simpler systems of ordinary differential equations.

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