Let $\gamma$ be a smooth regular curve $I \to \mathbb{R}^{3}$, and let $X \colon I \to \mathbb{R}^{3}$ be a smooth vector field along $\gamma$. Define $E$ to be a unit vector field along $\gamma$ such that $E(t) \times X(t) = 0$ for all $t \in I$.
I would like to understand under what conditions the vector field $E$ is well-defined, i.e., continuous. I guess that $E$ is continuous if every zero of $X$ is isolated. Am I correct?
Could one prove well-definedness of $E$ under any weaker assumption?
EDIT. As discussed in the comments, the vector field $E$ may fail to exist even if $X$ has isolated zeros. Does assuming that $X$ be analytic make any difference?