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Quadratic differential equations whose associated algebra has an automorphism of order two are studied. Under hypotheses that naturally generalize the cases where the even or odd part of the algebra is one dimensional, the following are examined: structure theory of the associated algebra (ideal structure, simplicity, solvability, and nilpotence), derivations and first integrals, trajectories given by derivations, and Floquet decompositions.

Quadratic differential equations in $\mathbb {Z}_2$-graded algebras