Need help understanding a lift of a vector field

This is a question from my differential geometry assignment:

Let $\pi:M\to N$ be a submersion between two smooth manifolds and $X\in \Gamma(TN)$ is a vector field. We need to show that there is a smooth vector field on $M$ that is $\pi$-related to $X$. Also determine a necessary and sufficient condition on $\pi$ for a lift of any vector field to be unique.

So what I’ve got so far is that, since $\pi$ is a submersion, at each $p\in M$, $\pi_{*p}$ is surjective. So for every $X_{\pi(p)}$ I can choose a $Y_p\in T_pM$ such that $\pi_{*p}Y_p=X_{\pi(p)}$ and define a local smooth vector field in a neighbourhood $U_p$ of $p$. Do it for every $p\in M$ and $\{U_p\}$ gives an open cover of $M$.

My problem here is that, firstly I don’t know how I could get a local smooth vector field from $Y_p$. I thought of using an integral curve and extend $Y_p$ according to the curve, but it does not give me a vector field on an open set.

Secondly, in order to get a global smooth vector field, in the overlapping, say a point $r\in U_p\cap U_q$, I am going to have $Y^1_r, Y^2_r$ that come from the two local vector fields defined by two points $p$ and $q$. Then I should probably sum up all the vectors in $T_rM$, then use the partition of unity subordinate to $\{U_p\}$. But I’m not sure how to apply the partition of unity.

I’m stuck now, how can I go from here? Please do not post complete solutions.


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