# Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras

## TL;DR

This paper introduces a new method for constructing superintegrable systems using block separation of variables, generalizing classical models and deriving their quadratic symmetry algebras with universal features.

## Contribution

The authors develop a novel approach for building superintegrable systems with arbitrary functions and singular terms, extending known models like the oscillator and Kepler systems.

## Key findings

- Derived exact energy spectra for new superintegrable systems.
- Constructed quadratic algebras for integrals of motion.
- Showed universality of quadratic algebra structures regardless of potential functions.

## Abstract

We present a new method for constructing $D$-dimensional minimally superintegrable systems based on block coordinate separation of variables. We give two new families of superintegrable systems with $N$ ($N\leq D$) singular terms of the partitioned coordinates and involving arbitrary functions. These Hamiltonians generalize the singular oscillator and Kepler systems. We derive their exact energy spectra via separation of variables. We also obtain the quadratic algebras satisfied by the integrals of motion of these models. We show how the quadratic symmetry algebras can be constructed by novel application of the gauge transformations from those of the non-partitioned cases. We demonstrate that these quadratic algebraic structures display an universal nature to the extent that their forms are independent of the functions in the singular potentials.