Suppose to consider two arbitrary smooth vector fields $X: M\rightarrow TM$ and $Y: M\rightarrow TM$ defined on the same differentiable manifold $M$. Is there always a smooth map $A: TM\rightarrow TM$ (or a composition of smooth maps, none of them necessarily induced by a diffeomorphism $\varphi: M\rightarrow M$), such that $Y=A(X)$?
If the answer is no, what are the conditions under which I can find such a smooth map $A$?
If you know the answer, could you please suggest me a reference where this problem is treated?
The vector space structure makes this quite simple: we can do this using a fiberwise translation of the tangent bundle. More explicitly, define $A: TM \to TM$ by $$A(V_p) = V_p + Y_p - X_p.$$ In standard coordinates $(x^i, dx^i)$ on the tangent bundle this is simply $A^i(p,V) = (p, V^i + Y^i - X^i)$, so it smooth by the smoothness of $X,Y$; and the desired property $A(X) = Y$ is clear.