  In this work we study Beltrami fields with non-constant proportionality
factor on $\mathbb{R}^3$. More precisely, we analyze the existence of vector
fields $X$ satisfying the equations $curl(X)=fX$ and $div(X)=0$ for a given
$f\in C^\infty(\mathbb R^3)$ in a neighborhood of a point $p\in\mathbb{R}^3$.
Since the regular case has been treated previously, we focus on the case where
$p$ is a non-degenerate critical point of $f$. We prove that for a generic
Morse function $f$, the only solution is the trivial one $X\equiv 0$ (here
generic refers to explicit arithmetic properties of the eigenvalues of the
Hessian of $f$ at $p$). Our results stem from the introduction of algebraic
obstructions, which are discussed in detail throughout the paper.
