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In this paper, we present a complete solution to the Cauchy dual subnormalityproblem for torally expansive toral $3$-isometric weighted $2$-shifts. Thissolution is obtained by solving a couple of Hausdorff moment problems arisingfrom $2$-variable polynomials of lower bi-degree.

A complete solution to the Cauchy dual subnormality problem for torally expansive toral $3$-isometric weighted $2$-shifts