On-flow and strong solutions to Killing-type equations

If we impose infinitesimal invariance up to a boundary term of the action functional for Lagrangian ordinary differential equations, we are led to Killing-type equations, which are related to first integrals through Noether theorem. We review the "on-flow" and "strong" interpretations of the Killing-type equation, and for each we detail the complete explicit structure of the solution set in terms of the associated first integral. We give examples that reappraise the usefulness of the "on-flow" solutions. Finally, we describe an equivalent alternative approach to variational invariance.