  The Berwald-Landsberg problem is considered for two dimensional manifolds. We
sketch the proof that there are not $\mathcal{C}^5$-regular $y$-global pure
Landsberg surfaces. The method used consists on considerer the holonomy
representation of the averaged Chern connection and then exhausting all the
possible cases, showing that in dimension 2 Landsberg condition implies Berwald
condition.
