  We show that there are not pure $\mathcal{C}^5$ regular y-global Landsberg
surfaced. The proof is based on the averaged connection associated with the
linear Chern's connection and the classification of irreducibles holonomies of
torsion-free affine connections. The structure consists on exausting all the
possible cases and showing that in dimension 2 Landsberg condition implies
Berwald condition.
