Branches of Markoff $m$-triples with two $k$-Fibonacci components

We study infinite paths of Markoff $m$-triples, that is, solutions to the generalised Markoff equation $$x^2+y^2+z^2=3xyz+m,$$ with $m>0$, with at least two $k$-Fibonacci components. First, we obtain a complete classification of Markoff $m$-triples whose last two entries are $k$-Fibonacci numbers and that are not roots of any Markoff trees. We then prove that every such infinite path is contained in a branch, starting at a triple of the form $$(\frac{F_k(4r)}{3F_k(2r)},F_k(\mathrm{\ell }+2r),F_k(\mathrm{\ell }+4r)),$$ where $r$ is an odd integer, $\mathrm{\ell }\in \{1,2,\dots ,2r\}$ and $3\nmid k$. These branches are distributed among exactly $2r$ distinct trees.