Holomorphic foliations of degree two and arbitrary dimension

Let $\mathcal{F}$ be a holomorphic foliation of degree $2$ on $\mathbb{P}^n$ with dimension $k\geq 2$. We prove that either $\mathcal{F}$ is algebraically integrable, or $\mathcal{F}$ is the linear pull-back of a degree-2 foliation by curves on $\mathbb{P}^{n-k+1}$, or $\mathcal{F}$ is a logarithmic foliation of type $(1^{n-k +1},2)$, or $\mathcal{F}$ is a logarithmic foliation of type $(1^{n-k+3})$, or $\mathcal{F}$ is the linear pull-back of a degree-2 foliation of dimension 2 on $\mathbb{P}^{n-k+2}$ tangent to an action of the Lie algebra $\mathfrak{aff}(\mathbb{C})$. As a byproduct, we describe the geometry of Poisson structures on $\mathbb{P}^n$ with generic rank two.
Comment: 19 pages; major changes with several improvements and corrections. Comments welcome!