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In this paper, we continue our investigation of the triple Hodge integralssatisfying the Calabi-Yau condition. For the tau-functions, which generatethese integrals, we derive the complete families of the Heisenberg-Virasoroconstraints. We also construct several equivalent versions of the cut-and-joinoperators. These operators describe the algebraic version of topologicalrecursion. For the specific values of parameters associated with the KdVreduction, we prove that these tau-functions are equal to the generatingfunctions of intersection numbers of $\psi$ and $\kappa$ classes. We interpretthis relation as a symplectic invariance of the Chekhov--Eynard--Orantintopological recursion and prove this recursion for the general $\Theta$-case.

KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions