This paper studies a family of nonlinear oscillators and uses them to produce two-dimensional geometric shapes (metrics) whose straightest paths, called geodesics, have extra conserved quantities. In plain terms, the authors find new examples of surfaces where the motion of a particle along geodesics is unusually simple to describe because the system has extra symmetries. Some of these metrics are “superintegrable,” meaning they have more conserved quantities than are typical, and for those the unparametrized geodesics can be written down explicitly.

The technical starting point is an autonomous cubic oscillator equation that represents the projection of two-dimensional geodesic motion. The researchers look for cases where this projected equation is both integrable (it has useful conserved quantities, or first integrals) and metrisable (it actually comes from some metric). Their main method is to intersect the conditions needed for integrability with the conditions needed for metrisability. They exploit classes of oscillators that can be linearized by nonlocal transformations, and these linearizability conditions split into three cases. Each case yields a family of integrable metrics that are parametrized by arbitrary functions. In the first case they obtain a family of superintegrable metrics (with a linear and a transcendental conserved quantity). In the other two cases they obtain new integrable families whose conserved quantities are typically transcendental functions.

A few short explanations of terms help to read the results. A metric is the mathematical object that measures distances on a surface. A geodesic is the analog of a straight line on that surface. A first integral is a function that stays constant along geodesics; having enough of them makes the motion integrable in the classical sense. “Metrisable” means the projected oscillator equation really comes from some metric; not every such equation does. Relative Killing vectors are a form of symmetry of the geodesic flow; the paper gives explicit formulas linking these symmetry vectors to invariants of the oscillator equation that are linear in the derivative.