In the 1990's, Henri Cohen asked whether the map $(x,y) \mapsto (\sqrt{1+x^2}-y,x)$ from $\mathbb{R}^2$ to itself is integrable.  In other words, are the orbits confined to the level curves of some nice function?  It certainly looks like they are if you make plots (the postscript file at <http://www.math.washington.edu/~cohn/cohen.ps> shows some orbits and is easy to modify), but no algebraic function could play this role (a result of Rychlik and Torgerson; <http://nyjm.albany.edu/j/1998/4-5.html>).  Years ago, someone told me it was unlikely that it was really integrable, because careful investigation identified hyperbolic periodic orbits, but I never learned more and I didn't see this in my crude plots.  Unfortunately, I don't remember who told me.

Is the map actually integrable?  This may well be an open problem: I looked at the papers that cite the Rychlik and Torgerson paper, except for one I couldn't access, but I found no evidence that this question has been resolved.  On the other hand, there don't seem to be many papers on this topic, and perhaps that's because the conjecture turned out to be false, with a disproof that was never published.

If it isn't integrable, then what could explain the near integrability?  From my naive perspective, a simple system like this that looks integrable but isn't would be really amazing.  Are such things more common than I realize?