Towards construction of superintegrable basis in matrix models

We develop methods for systematic construction of superintegrable polynomials in matrix/eigenvalue models. Our consideration is based on a tight connection of superintegrable property of Gaussian Hermitian model and W1+∞ algebra in Fock representation. Motivated by this example, we propose a set of assumptions that may allow one to recover superintegrable polynomials. The two main assumptions are the box adding/removing rule (Pierri rule) and the existence of Hamiltonian for superintegrable polynomials. We describe our method in detail in case of the Gaussian Hermitian model, and then apply it to the cubic Kontsevich model.