  In this paper, we continue our investigation of the triple Hodge integrals
satisfying the Calabi-Yau condition. For the tau-functions, which generate
these integrals, we derive the complete families of the Heisenberg-Virasoro
constraints. We also construct several equivalent versions of the cut-and-join
operators. These operators describe the algebraic version of topological
recursion. For the specific values of parameters associated with the KdV
reduction, we prove that these tau-functions are equal to the generating
functions of intersection numbers of $\psi$ and $\kappa$ classes. We interpret
this relation as a symplectic invariance of the Chekhov--Eynard--Orantin
topological recursion and prove this recursion for the general $\Theta$-case.
