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.