  We study the correlators $W_{g,n}$ arising from Orlov-Scherbin 2-Toda tau
functions with rational content-weight $G(z)$, at arbitrary values of the two
sets of time parameters. Combinatorially, they correspond to generating
functions of weighted Hurwitz numbers and $(m,r)$-factorisations of
permutations. When the weight function is polynomial, they are generating
functions of constellations on surfaces in which two full sets of degrees
(black/white) are entirely controlled, and in which internal faces are allowed
in 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 the
2-matrix model which corresponds to $G(z)=1+z$, by the addition of arbitrarily
many free parameters. Our method relies both on the Albenque-Bouttier
combinatorial proof of Eynard's result by slice decompositions, which is strong
enough 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 no
internal faces) by work of Bychkov-Dunin-Barkowski-Kazarian-Shadrin (or
Alexandrov-Chapuy-Eynard-Harnad for the polynomial case), and on the general
idea of deformation of spectral curves due to Eynard and Orantin, which we make
explicit in this case. As a result of TR, we obtain strong structure results
for all fixed-genus generating functions.
