2020/02/24 by Galliano Valent, Galliano, VALENT
Chemistry · Decision Sciences · Physics and Astronomy · #32C05 #37E99 #37K25 #Atomic and Molecular Physics #Chemical Thermodynamics and Molecular Structure #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #S1V99 #Scientific Measurement and Uncertainty Evaluation
paper · pdf · doi:10.48550/arxiv.2002.10507
openalex publication_date 2020/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Koenigs constructed a family of two dimensional superintegrable (SI) models with one linear and two quadratic integrals in the momenta, shortly (1,2). More recently Matveev and Shevchishin have shown that this construction does generalize to models with one linear and two cubic integrals i.e. (1,3), up to the solution of a non-linear ordinary differential equation. Our explicit solution of this equation allowed for the construction of these SI systems and led to the proof that the systems globally defined on S2 are Zoll. We will generalize these results to the case (1,n) for any n ≥ 2. Our approach is again constructive and shows the existence, when n is odd, of metrics globally defined on S2 which are indeed Zoll (under appropriate restrictions on the parameters), while if n is even the metrics we found are never globally defined on S2, as it is already the case for the (1,2) models constructed by Koenigs.