Markoff-Fibonacci m-triples

We classify all solution triples with Fibonacci components to the equation a²+b²+c²=3abc+m, for positive m. We show that for $m=2$ they are precisely $(1,F(b),F(b+2))$, with even b; for $m=21$, there exist exactly two Fibonacci solutions $(1,2,8)$ and $(2,2,13)$ and for any other m there exists at most one Fibonacci solution, which, in case it exists, is always minimal (i.e. it is a root of a Markoff tree). Moreover, we show that there is an infinite number of values of m admitting exactly one such solution.