Isospin particle systems on quaternionic projective spaces

Abstract

We construct the isospin particle system on $n$-dimensional quaternionic projective spaces in the presence of the Belavin-Polyakov-Schwarz-Tyupkin instanton by the reduction from the free particle on $(2n+1)$-dimensional complex projective space. Then we add to this system a ``quaternionic oscillator potential'' and show that this oscillatorlike system is superintegrable. We show that, besides the analogs of quadratic constants of motion of the spherical (Higgs) and $\mathbb{C}{P}^{n}$ oscillators, it possesses the third-order constants of motion, which are functionally independent from the quadratic ones.