A Sylvester-Gallai configuration in the the complex projective plane is a finite number of $n\ge 2$ points in the complex projective plane such that there is no line through exactly two of them. Trivial examples are obtained by taking $n\ge 3$ points on the same line. There is also the classical Hessian configuration, obtained by taking the nine inflexion points of a smooth cubic.

Question.Is there any other example? Is there a way to classify these examples? I would be particularly interested in finding a non-collinear example with an even number of points. Does it exist?

Remark: In higher dimension, it is known that the configuration of points has to be coplanar. If the points have coordinates defined over $\mathbb{R}$, the Sylvester-Gallai theorem shows that any configuration as above is in fact collinear. Over finite fields one can of course find plenty of configurations by taking all points.