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In this paper, we study left invariant conic Finsler metrics on the2-dimensional non-Abelian Lie group $G$ with nowhere vanishing spray vectorfields, and classify those satisfying the constant curvature condition, theLandsberg condition or the Berwald condition respectively. We prove that anyleft invariant conic Landsberg metric on $G$ must be Berwald. This discoveryenable us to propose a homogeneous conic Landsberg Conjecture, which guessesthat every homogeneous conic Landsberg metric is Berwald, and prove the2-dimensional case for it.

Special left invariant conic Finsler metrics and homogeneous conic Landsberg Problem in two dimension