  We investigate the local metrizability of Finsler spaces with $m$-Kropina
metric $F = \alpha^{1+m}\beta^{-m}$, where $\beta$ is a closed null 1-form. We
show that such a space is of Berwald type if and only if the
(pseudo-)Riemannian metric $\alpha$ and 1-form $\beta$ have a very specific
form in certain coordinates. In particular, when the signature of $\alpha$ is
Lorentzian, $\alpha$ belongs to a certain subclass of the Kundt class and
$\beta$ generates the corresponding null congruence, and this generalizes in a
natural way to arbitrary signature. We use this result to prove that the affine
connection on such an $m$-Kropina space is locally metrizable by a
(pseudo-)Riemannian metric if and only if the Ricci tensor constructed form the
affine connection is symmetric. In particular we construct all counterexamples
of this type to Szabo's metrization theorem, which has only been proven for
positive definite Finsler metrics that are regular on all of the slit tangent
bundle.
