In the present paper, we find out necessary and sufficient conditions for a
Finsler surface (M,F) to be Landsbregian in terms of the Berwald curvature
2-forms. We study Finsler surfaces which satisfy some flag curvature K
conditions, viz., V(K)=0,V(K)=βI/F2 and
V(K)=βIK, where I is the Cartan scalar. In order to
do so, we investigate some geometric objects associated with the global Berwald
distribution D:=span{S,H,V:=JH} of a
2-dimensional Finsler metrizable nonflat spray S. We obtain some
classifications of such surfaces and show that under what hypothesis these
surfaces turn to be Riemannian. The existence of a first integral for the
geodesic flow in each case has some remarkable consequences concerning rigidity
results. We prove that a Finsler surface with V(K)=βI/F2 and
either S(K)=0 or S(J)=0 is Riemannian. Further, a Finsler surface
with V(K)=βIK and S(K)=0 is Riemannian.Comment: 10 page