The aim of this paper is to introduce and study the class of conics provided by the symmetric matrices of the adjoint representation of the Lie group $SU(2)=S^3$. This class depends on three real parameters as components of a point of sphere $S^2$ and various relationships between these parameters give special subclasses of conics. A symmetric matrix inspired by one giving by Barning as Pythagorean triple preserving matrix and associated hyperbola are carefully analyzed. We extend this latter hyperbola to a class of hyperbolas with integral coefficients. A complex approach is also included.