"Given the contravariant vector field $V{\mu}(x)$, we cosider te differential equation $$\frac{dx^{\mu}}{d\lambda}=V^{\mu}(x).$$ The solution $x^{\mu}$ is a map from $\mathbb{R}\rightarrow M$ is called an integral curve. There is an integral curve through every point of any open subset of $M$ in which the vector field does not vanish." Quote from Supergravity book, but the essence is mathematical.
My question is what happens if the field vanishes? I mean why wouldn't an integral curve exist if the vector field vanishes?