In this work we study Beltrami fields with non-constant proportionality factor on R3. More precisely, we analyze the existence of vector fields X satisfying the equations curl(X)=fX and div(X)=0 for a given f∈C∞(R3) in a neighborhood of a point p∈R3. 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≡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.