I am aware of the following examples of normal surfaces in $\mathbb P^3$ that are projectively isomorphic to their dual varieties:


1. the smooth quadric;
2. Kummer surfaces;
3. The surface with the equation $x\_0^3=x\_1x\_2x\_3$ (in homogeneous coordinates).


What else is known? The base field is algebraically closed of characteristic zero.


Thank you in advance,
Serge



Self-dual surfaces in $\mathbb P^3$ with isolated singularities
In his paper [Some invariants for conics and their applications, *Publ. RIMS (Kyoto Univ.)* **19** (1983), 1139-1151] Naruki gives an example of a self-dual quartic surface in $\mathbb{P}^3$ with three singular points of type $A\_3$ and seven points of type $A\_1$ (i.e., ordinary double points). 


The paper can be dowloaded [here](http://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=22&ved=0CDoQFjABOBQ&url=http%3A%2F%2Fwww.ems-ph.org%2Fjournals%2Fshow_pdf.php%3Fissn%3D0034-5318%26vol%3D19%26iss%3D3%26rank%3D10&ei=tRmHUr67ENT54QS0o4B4&usg=AFQjCNGzATXePi3bFar7IJqVykg82eoigA&bvm=bv.56643336,d.bGE&cad=rja). 


