### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 008, 51 pages      arXiv:1704.05805      https://doi.org/10.3842/SIGMA.2018.008
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

### Darboux Integrability of Trapezoidal $H^{4}$ and $H^{6}$ Families of Lattice Equations II: General Solutions

Giorgio Gubbiotti ab, Christian Scimiterna b and Ravil I. Yamilov c
a) School of Mathematics and Statistics, F07, The University of Sydney, New South Wales 2006, Australia
b) Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre and Sezione INFN di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
c) Institute of Mathematics, Ufa Scientific Center, Russian Academy of Sciences, 112 Chernyshevsky Str., Ufa 450008, Russia

Received April 26, 2017, in final form January 16, 2018; Published online February 02, 2018

Abstract
In this paper we construct the general solutions of two families of quad-equations, namely the trapezoidal $H^{4}$ equations and the $H^{6}$ equations. These solutions are obtained exploiting the properties of the first integrals in the Darboux sense, which were derived in [Gubbiotti G., Yamilov R.I., J. Phys. A: Math. Theor. 50 (2017), 345205, 26 pages]. These first integrals are used to reduce the problem to the solution of some linear or linearizable non-autonomous ordinary difference equations which can be formally solved.

Key words: quad-equations; Darboux integrability; exact solutions; CAC.

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