The Lie commutators on the space of the smooth functions from $R^1$ to $R^2$

In this paper we consider the classification problem of the local algebras Lie on the space $C^\infty(R^n,R^m)$. For $n=1$, $m=2$ and the symmetry analytical Lie commutators of the first order we obtain full classification under the module of the action of the group $GL_2(F)$, where $F$ is the space analytical functions from one variable.