The S-function at arbitrary order: hyperplanes of symmetries and the realisability condition

Abstract

For a scalar equation $y^{(n)}=\phi (x,y,\dots ,y^{(n-1)})$ the $S$-function of the extended Prelle--Singer method is a vector $S=(S_0,\dots ,S_{n-2})$, the null-form vector of Chandrasekar, Senthilvelan, Lakshmanan and collaborators, and the jet gradient of a first integral solves the adjoint of the variational equation. We describe the solutions of the system satisfied by $S$ intrinsically, and determine which of them come from first integrals. Over the field $C$ of first integrals the solutions are exactly the hyperplanes of the $n$-dimensional space $VV$ of symmetry characteristics, recovered from $n-1$ independent characteristics by a Plücker formula; in flow-box coordinates they are the hyperplane fields on the local space of leaves. A solution is the $S$-vector of a first integral if and only if the corresponding hyperplane field is integrable. This Frobenius condition splits into the $S$-system itself plus $\left(\genfrac{}{}{0ex}{}{n}{3}\right)$ further differential conditions, so it is vacuous for $n=2$ and a genuine restriction for every $n\ge 3$, for which we give an explicit non-realisable solution. The same picture explains why $\lambda$-symmetries and $S$-functions coincide only when $n=2$: line fields and hyperplane fields agree only on surfaces. For the computation of $S$ we record that the system is equivalent to one scalar equation for the last component, which is the Riccati form of the adjoint variational equation, and we spell out what this does and does not imply for a rational search.