  We derive and study Yangian Ward identities for the infinite class of
four-point ladder integrals and their Basso-Dixon generalisations. These
symmetry equations follow from interpreting the respective Feynman integrals as
correlation functions in the bi-scalar fishnet theory. Alternatively, the
presented identities can be understood as anomaly equations for a momentum
space conformal symmetry. The Ward identities take the form of inhomogeneous
extensions of the partial differential equations defining the Appell
hypergeometric functions. We employ a manifestly conformal tensor reduction in
order to express these inhomogeneities in compact form, which are given by
linear combinations of Basso-Dixon integrals with shifted dimensions and
propagator powers. The Ward identities naturally generalise to a one-parameter
family of D-dimensional integrals representing correlators in the generalised
fishnet theory of Kazakov and Olivucci. When specified to two spacetime
dimensions, the Yangian Ward identities decouple. Using separation of
variables, we explicitly bootstrap the solution for the conformal 2D box
integral. The result is a linear combination of Yangian invariant products of
Legendre functions, which reduce to elliptic K integrals for an isotropic
choice of propagator powers. We comment on differences in the transcendentality
patterns in two and four dimensions and their relations to discontinuities.
