Results and Conjectures about Order [inline-graphic not available: see fulltext] Lyness' Difference Equation [inline-graphic not available: see fulltext] in [inline-graphic not available: see fulltext], with a Particular Study of the Case [inline-graphic not available: see fulltext]

We study order [inline-graphic not available: see fulltext] Lyness' difference equation in [inline-graphic not available: see fulltext], with [inline-graphic not available: see fulltext] and the associated dynamical system [inline-graphic not available: see fulltext] in [inline-graphic not available: see fulltext]. We study its solutions (divergence, permanency, local stability of the equilibrium). We prove some results, about the first three invariant functions and the topological nature of the corresponding invariant sets, about the differential at the equilibrium, about the role of 2-periodic points when [inline-graphic not available: see fulltext] is odd, about the nonexistence of some minimal periods, and so forth[inline-graphic not available: see fulltext] and discuss some problems, related to the search of common period to all solutions, or to the second and third invariants. We look at the case [inline-graphic not available: see fulltext] with new methods using new invariants for the map [inline-graphic not available: see fulltext] and state some conjectures on the associated dynamical system in [inline-graphic not available: see fulltext] in more general cases.