  We present a basis of eigenvectors for the graph building operators acting
along the mirror channel of planar fishnet Feynman integrals in $d$-dimensions.
The eigenvectors of a fishnet lattice of length $L$ depend on a set of $L$
quantum numbers $(u_k,l_k)$, each associated with the rapidity and bound-state
index of a lattice excitation. Each excitation is a particle in
$(1+1)$-dimensions with $O(d)$ internal symmetry, and the wave-functions are
formally constructed with a set of creation/annihilation operators that satisfy
the corresponding Zamolodchikovs-Faddeev algebra. These properties are proved
via the representation - new to our knowledge - of the matrix elements of the
fused R-matrix with $O(d)$ symmetry as integral operators on the functions of
two spacetime points. The spectral decomposition of a fishnet integral we
achieved can be applied to the computation of Basso-Dixon integrals in higher
dimensions.
