The Lipman-Zariski conjecture in genus one higher

We prove the Lipman-Zariski conjecture for complex surface singularities with $p_g-g-b\le 2$. Here $p_g$ is the geometric genus, $g$ is the sum of the genera of the exceptional curves and $b$ is the first Betti number of the dual graph. This improves on a previous result of the second author. As an application, we show that a compact complex surface with locally free tangent sheaf is smooth as soon as it admits two generically linearly independent twisted vector fields and its canonical sheaf has at most two global sections.