Suppose we have a smooth manifold $ M$ and the tangent space of every point $ x \in M$ has non-empty intersection with a given linear subspace. Could we find a curve on $ M$ such that the tangent vector of point on thus curve always belongs to this linear subspace?

$ \textbf{My attempt:}$ If the dimension for the linear subspace is one or two, I think I can find the curve. But I don’t know if the dimension is bigger than two?

I will appreciate for any useful answers and comments