TITLE: Self-dual surfaces in $\mathbb P^3$ with isolated singularities
QUESTION [5 upvotes]: I am aware of the following examples of normal surfaces in $\mathbb P^3$ that are projectively isomorphic to their dual varieties:

the smooth quadric;
Kummer surfaces;
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

REPLY [6 votes]: 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.