Binary differential equations with symmetries

Abstract

This paper introduces the study of occurrence of symmetries in binary differential equations (BDEs). These are implicit differential equations given by the zeros of a quadratic 1-form, \begin{document}$ a(x,y)dy^2 + b(x,y)dxdy + c(x,y)dx^2 = 0, $\end{document} for \begin{document}$ a, b, c $\end{document} smooth real functions defined on an open set of \begin{document}$ \mathbb{R}^2 $\end{document} . Generically, solutions of a BDE are given as leaves of a pair of foliations, and the action of a symmetry must depend not only whether it preserves or inverts the plane orientation, but also whether it preserves or interchanges the foliations. The first main result reveals this dependence, which is given algebraically by a formula relating three group homomorphisms defined on the symmetry group of the BDE. The second main result adapts methods from invariant theory of compact Lie groups to obtain an algorithm to compute general expressions of equivariant quadratic 1-forms under each compact subgroup of the orthogonal group \begin{document}$ {{\bf{O}}(2)} $\end{document} .