Reconstructing a Lattice Equation: a Non-Autonomous Approach to the Hietarinta Equation

In this paper we construct a non-autonomous version of the Hietarinta equation [J. Hietarinta. A new two-dimensional lattice model that is consistent around the cube. \emph{J. Phys. A: Math. Gen.}, 37:L67--L73, 2004] and study its integrability properties. We show that this equation possess linear growth of the degrees of iterates, generalized symmetries depending on arbitrary functions, and consequently that is Darboux integrable. We use the first integrals to provide a general solution of this equation. Finally we show that this equation is a particular case of the non-autonomous $Q_\text{V}$ equation, and we provide a non-autonomous M\"obius transformation to another equation found in [J. Hietarinta. Searching for CAC-maps. \emph{J. Nonlinear Math. Phys.}, 12:223--230, 2005].