This paper studies a family of nonlinear oscillators and shows how they produce new two‑dimensional geometric metrics whose geodesic motion is highly regular. The oscillators are the autonomous (time‑independent) case of a two‑dimensional projective connection. “Metrisable” here means that the projected equation for curves comes from geodesics of some Riemannian metric. “Integrable” means there are conserved quantities, or “first integrals,” that make the motion solvable in a classical sense; “superintegrable” means there are more conserved quantities than the number of degrees of freedom.

Concretely, the authors study cubic oscillator equations of the form y'' + k(y)(y')^3 + h(y)(y')^2 + f(y)y' + g(y) = 0, where primes mean d/dx and the coefficient functions depend on y alone. These equations define a vector field X = u ∂y − (k u^3 + h u^2 + f u + g) ∂u with u = y'. To decide when such an equation is metrisable they use a known linear overdetermined system (a form of Liouville’s conditions) for three auxiliary functions ψ1, ψ2, ψ3. A solution of that system with nonzero determinant gives a metric whose unparameterized geodesics are the equation’s solutions.

Using this framework, the authors construct several classes of oscillators that are simultaneously integrable and metrisable. That construction yields families of two‑dimensional metrics that are integrable or superintegrable and that are parametrized by arbitrary functions. In the superintegrable family they obtain an explicit formula for the unparameterized geodesics. In the integrable families they find two new classes of metrics whose additional conserved quantities are transcendental functions rather than simple polynomials or rational functions.

Methodologically they take a novel route: instead of guessing metrics, they look for overlaps between integrability and metrisability conditions for the projective equations. They allow nonlocal transformations that preserve the equation’s form and integrability but not necessarily the metric form. The linearizability conditions under these generalized transformations split into three cases. Each case produces a family of integrable metrics, and the authors classify the dimensions of the associated projective Lie algebra (the algebra of point symmetries of the projective structure). For certain parameter values they recover previously known cases: the symmetry algebra is either sl(2,R) or sl(3,R). Those two outcomes correspond respectively to metrics that are Darboux‑superintegrable (having four independent second‑degree polynomial conserved quantities) or to flat metrics of constant curvature.