  Lindstr\"om theorem obviously fails as a characterization of
$\mathcal{L}_{\omega \omega}^{-} $, first-order logic without identity. In this
note we provide a fix: we show that $\mathcal{L}_{\omega \omega}^{-} $ is
\emph{maximal} among abstract logics satisfying a weak form of the isomorphism
property (suitable for identity-free languages and studied in \cite{Casa}), the
L\"owenheim--Skolem property, and compactness. Furthermore, we show that
compactness can be replaced by being recursively enumerable for validity under
certain conditions. In the proofs we use a form of strong upwards
L\"owenheim--Skolem theorem not available in the framework with identity.
