  A holomorphic pre-foliation $\mathscr{F}=\ell\boxtimes\mathcal{F}$ of
co-degree $1$ and degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is the data of a
line $\ell$ of $\mathbb{P}^{2}_{\mathbb{C}}$ and a holomorphic foliation
$\mathcal{F}$ on $\mathbb{P }^{2}_{\mathbb{C}}$ of degree $d-1.$ We study
pre-foliations of co-degree $1$ on $\mathbb{P}^{2}_{\mathbb{ C}}$ with a flat
Legendre transform (dual web). After having established some general results on
the flatness of the dual $d$-web of a homogeneous pre-foliation of co-degree
$1$ and degree $d$, we describe some explicit examples and we show that up to
automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$ there are two families and six
examples of homogeneous pre-foliations of co-degree $1$ and degree $3$ on
$\mathbb {P}^{2}_{\mathbb{C}}$ with a flat dual web. This allows us to prove an
analogue for pre-foliations of co-degree $1$ and degree~$3$ of a result,
obtained in collaboration with D. Mar\'{\i}n, on foliations of degree $3$ with
non-degenerate singularities and a flat Legendre transform. We also show that
the dual web of a reduced convex pre-foliation of co-degree $1$ on
$\mathbb{P}^{2}_{\mathbb{C}}$ is flat. This is an analogue of a result on
foliations of $\mathbb{P}^{2}_{\mathbb{C}}$ due to D. Mar\'{\i}n and J. V.
Pereira.
