Deformation of superintegrability in the Miwa-deformed Gaussian matrix model

We consider an arbitrary deformation of the Gaussian matrix model parametrized by Miwa variables <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>z</a:mi><a:mi>a</a:mi></a:msub></a:math>. 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 polynomial made from the values of skew Schur functions at the Gaussian locus <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>p</c:mi><c:mi>k</c:mi></c:msub><c:mo>=</c:mo><c:msub><c:mi>δ</c:mi><c:mrow><c:mi>k</c:mi><c:mo>,</c:mo><c:mn>2</c:mn></c:mrow></c:msub></c:math>. This linear operation includes multiplication with an exponential <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>e</e:mi><e:mrow><e:msubsup><e:mi>z</e:mi><e:mi>a</e:mi><e:mn>2</e:mn></e:msubsup><e:mo>/</e:mo><e:mn>2</e:mn></e:mrow></e:msup></e:math> 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>K</g:mi></g:math>-polynomials, which appeared in superintegrable correlators at the Gaussian point (strict superintegrability). We also consider in great detail the generating function of correlators <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:mo stretchy="false">⟨</i:mo><i:mo stretchy="false">(</i:mo><i:mi>Tr</i:mi><i:mi>X</i:mi><i:msup><i:mrow><i:mo stretchy="false">)</i:mo></i:mrow><i:mrow><i:mi>k</i:mi></i:mrow></i:msup><i:mo stretchy="false">⟩</i:mo></i:mrow></i:math> in this model, and discuss its integrable determinant representation. At last, we describe deformation of all results to the Gaussian <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>β</o:mi></o:math>-ensemble.