There are no $\mathcal{C}^5$-Regular Pure $y$-Global Landsberg Surfaces
  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. 
Ricardo Gallego Torrome