Deformation of superintegrability in the Miwa-deformed Gaussian matrix model

We consider an arbitrary deformation of the Gaussian matrix model parameterized by Miwa variables z_a. One can look at it as a mixture of the Gaussian and logarithmic (Selberg) potentials, which are both superintegrable. The mixture is not, still one can find an explicit expression for an arbitrary Schur average as a linear transform of a finite degree polynomial made from the values of skew Schur functions at the Gaussian locus p_k=δ_k,2. This linear operation includes multiplication with an exponential e^z_a^2/2 and a kind of Borel transform of the resulting product, which we call multiple and enhanced. The existence of such remarkable formulas appears intimately related to the theory of auxiliary K-polynomials, which appeared in bilinear superintegrable correlators at the Gaussian point (strict superintegrability). We also consider in the very detail the generating function of correlators <( X)^k> in this model, and discuss its integrable determinant representation. At last, we describe deformation of all results to the Gaussian β-ensemble.