  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.

Algebraic Geometry;Commutative Algebra;Complex Variables