Darboux integrability of trapezoidal H 4 and H 4 families of lattice equations I: First integrals

In this paper we prove that the trapezoidal H4 and the H6 families of quadequations are Darboux integrable by constructing their first integrals. This result explains why the rate of growth of the degrees of the iterates of these equations is linear (Gubbiotti et al 2016 J. Nonlinear Math. Phys. 23 507-43), which according to the algebraic entropy conjecture implies linearizability. We conclude by showing how first integrals can be used to obtain general solutions.