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

## TL;DR

This paper develops a non-autonomous version of the Hietarinta lattice equation, demonstrating its integrability, deriving solutions, and connecting it to known equations in the literature.

## Contribution

It introduces a non-autonomous form of the Hietarinta equation and analyzes its integrability, symmetries, and solutions, linking it to the non-autonomous QV equation and other classifications.

## Key findings

- The non-autonomous Hietarinta equation exhibits linear degree growth.
- It possesses generalized symmetries depending on arbitrary functions.
- The equation is Darboux integrable and solvable via first integrals.

## Abstract

In this paper we construct a non-autonomous version of the Hietarinta equation [Hietarinta J., J. Phys. A: Math. Gen. 37 (2004), L67-L73] 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 that it is Darboux integrable. We use the first integrals to provide a general solution of this equation. In particular we show that this equation is a sub-case of the non-autonomous $Q_{\rm V}$ equation, and we provide a non-autonomous M\"obius transformation to another equation found in [Hietarinta J., J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 223-230] and appearing also in Boll's classification [Boll R., Ph.D. Thesis, Technische Universit\"at Berlin, 2012].