If $I$ is an interval of $\mathbb { R }$, we say that $u \in \mathcal { C } ^ { 1 } ( I , \mathbb { R } )$ satisfies (II.1) on $I$ if and only if
$$\forall t \in I , \quad u ( t ) \left( u ( t ) + 2 t u ^ { \prime } ( t ) \right) = - 1$$
A function $f \in \mathcal{C}^2(\mathbb{R}^2, \mathbb{R})$ satisfies (1) if and only if
$$\forall ( x , y ) \in \mathbb{R}^2, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$
Let $J$ be a non-empty open interval of $\mathbb { R }$, $\Omega ( J ) = \left\{ ( x , y ) \in \mathbb { R } ^ { 2 } , x y \in J \right\}$, $w$ in $\mathcal { C } ^ { 2 } ( J , \mathbb { R } )$ and $W$ the function defined by
$$\forall ( x , y ) \in \Omega ( J ) , \quad W ( x , y ) = w ( x y )$$

\begin{enumerate}
\item[II.D.1)] Show that $\Omega ( J )$ is a non-empty open set.
\item[II.D.2)] Show that $W$ is in $\mathcal { C } ^ { 2 } ( \Omega ( J ) , \mathbb { R } )$ and that there is equivalence between
\begin{enumerate}
\item[i.] $W$ satisfies (1) on $\Omega ( J )$,
\item[ii.] $w ^ { \prime }$ satisfies (II.1) on $J$.
\end{enumerate}
\item[II.D.3)] Show that $W$ is the restriction to $\Omega ( J )$ of a function in $\mathcal { P } _ { 2 }$ if and only if $w$ is affine.
\end{enumerate}