  In this paper we study a wide class of planar single-trace four point
correlators in the chiral conformal field theory ($\chi$CFT$_4$) arising as a
double scaling limit of the $\gamma$-deformed $\mathcal{N}=4$ SYM theory. In
the planar (t'Hooft) limit, each of such correlators is described by a single
Feynman integral having the bulk topology of a square lattice "fishnet" and/or
of an honeycomb lattice of Yukawa vertices. The computation of this class of
Feynmann integrals at any loop is achieved by means of an exactly-solvable spin
chain magnet with $SO(1,5)$ symmetry. In this paper we explain in detail the
solution of the magnet model as presented in our recent letter and we obtain a
general formula for the representation of the Feynman integrals over the
spectrum of the separated variables of the magnet, for any number of scalar and
fermionic fields in the corresponding correlator. For the particular choice of
scalar fields only, our formula reproduces the conjecture of B. Basso and L.
Dixon for the fishnet integrals.
