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We study the correlators $W_{g,n}$ arising from Orlov-Scherbin 2-Toda taufunctions with rational content-weight $G(z)$, at arbitrary values of the twosets of time parameters. Combinatorially, they correspond to generatingfunctions of weighted Hurwitz numbers and $(m,r)$-factorisations ofpermutations. When the weight function is polynomial, they are generatingfunctions of constellations on surfaces in which two full sets of degrees(black/white) are entirely controlled, and in which internal faces are allowedin addition to boundaries.  We give the spectral curve (the "disk" function $W_{0,1}$, and the "cylinder"function $W_{0,2}$) for this model, generalising Eynard's solution of the2-matrix model which corresponds to $G(z)=1+z$, by the addition of arbitrarilymany free parameters. Our method relies both on the Albenque-Bouttiercombinatorial proof of Eynard's result by slice decompositions, which is strongenough to handle the polynomial case, and on algebraic arguments.  Building on this, we establish the topological recursion (TR) for the model.Our proof relies on the fact that TR is already known at time zero (or,combinatorially, when the underlying graphs have only boundaries, and nointernal faces) by work of Bychkov-Dunin-Barkowski-Kazarian-Shadrin (orAlexandrov-Chapuy-Eynard-Harnad for the polynomial case), and on the generalidea of deformation of spectral curves due to Eynard and Orantin, which we makeexplicit in this case. As a result of TR, we obtain strong structure resultsfor all fixed-genus generating functions.

Topological recursion for Orlov-Scherbin tau functions, and constellations with internal faces