KP integrability of triple Hodge integrals, I. From Givental group to hierarchy symmetries

In this paper, we investigate a relation between the Givental group of rank one and Heisenberg-Virasoro symmetry KP hierarchy. We prove, that only two-parameter family operators can be identified with elements group. This describes triple Hodge integrals satisfying Calabi-Yau condition. Using identification two groups prove generating function condition its $\Theta$-version are tau-functions generalizes result Kazarian on integrability in case linear integrals.