Let $X$ and $Y$ be two smooth vector fields defined in a smooth $n$-dimensional manifold $M$, with $X$ and $Y$ never vanishing simultaneously. Say $\gamma:[0,\varepsilon_1]\to M$ is an integral curve of $X$ and $\rho:[0,\varepsilon_2]\to M$ is an integral curve of $Y$, with $\gamma(\varepsilon_1)=\rho(0)$.
Can I construct two smooth functions $\alpha,\beta \in C^{\infty}(M)$ such that the vector field $\alpha X+\beta Y$ has an integral curve that joins $\gamma(0)$ and $\rho(\varepsilon_2)$?
The problem, of course, is with the point of intersection $\gamma(\varepsilon_1)=\rho(0)$. I tried putting a coordinate patch around that point and using the tubular flow theorem to glue the two curves, but couldn't make it work.
Also, is it possible to do it without both functions ($\alpha$ and $\beta$) vanishing simultaneously?
I appreciate any suggestions!