  The set $\mathscr{F}(2;2)$ of quadratic foliations on the complex projective
plane can be identified with a \textsc{Zariski}'s open set of a projective
space of dimension 14 on which acts
  $\mathrm{Aut}(\mathbb{P}^2(\mathbb{C})).$ We classify, up to automorphisms of
$\mathbb{P}^2(\mathbb{C}),$ quadratic foliations with only one singularity.
There are only four such foliations up to conjugacy; whereas three of them have
a dynamic which can be easily described the dynamic of the fourth is still
mysterious. This classification also allows us to describe the action of
$\mathrm{Aut}(\mathbb{P}^2(\mathbb{C}))$ on $\mathscr{F}(2;2).$ On the one hand
we show that the dimension of the orbits is more than 6 and that there are
exactly two orbits of dimension $6;$ on the other hand we obtain that the
closure of the generic orbit in $\mathscr{F} (2;2)$ contains at least seven
orbits of dimension~7 and exactly one orbit of dimension $6.$
