Second-order superintegrable systems from semi-simple and nilpotent Frobenius structures

Recently, it was shown that a rich class of second-order (maximally) superintegrable systems has an underpinning Hesse-Frobenius structure, i.e.\ a Frobenius structure that is compatible with a Hessian structure such that the Hessian pre-potential is also a Frobenius pre-potential. Hence, these superintegrable systems arise, locally, from (possibly non-unital) Frobenius algebras. We use a conification to lift systems of non-zero constant sectional curvature to flat ones and we employ a direct product construction to generate higher-dimensional second-order maximally superintegrable systems on pseudo-Euclidean spaces. We apply the method to very basic semi-simple and nilpotent algebras and we explicitly construct the arising second-order superintegrable systems. All non-degenerate second-order maximally superintegrable systems in three dimensions arise from these examples.