Let $X$ be a vector field in $S^2$ that is $(ax_1+bx_2) \frac \partial {\partial x_1}+(cx_1+dx_2) \frac \partial {\partial x_2}$ on every point on the sphere except the north pole and $0$ on the north pole. $φ=(x_1,x_2)$ is the chart with formula $φ(x,y,z)=( \frac x{1-z}, \frac y{1-z})$.
I need to find $a,b,c,d \in \Bbb R$ so that $X$ is differentiable.
By changing coordinates from $(x_1,x_2)$ to $(y_1,y_2)= ( \frac x{1+z}, \frac y{1+z})$, it is easy to see that for $a=d$ and $b=-c$, $X$ is differentiable in (0,0,1) (the north pole).
My question is if these are the only solutions?