Isospin particle systems on quaternionic projective spaces

We construct the isospin particle system on $n$-dimensional quaternionic projective spaces in the presence of BPST-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 oscillator-like 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.