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This paper is devoted to the phenomenon of superintegrability. Thisphenomenon is manifested in the existence of a formula for character averages,expressed through the same characters at special points and of its variousgeneralization. In this paper we develop a method of proving such formulas fromfirst principle from Virasoro constraints and $W$-representation. We apply itto prove the formula for the Jack functions averages - appropriate analogue ofcharacters for the $\beta$-deformed Hermitian Gaussian matrix model. We alsosketch the construction of $W$-operators from Calogero-RuijsenaarsHamiltonians.

Superintegrability in $\beta$-deformed Gaussian Hermitian matrix model from $W$-operators