  A complex smooth prime Fano threefold $X$ of genus $9$ is related via
projective duality to a quartic plane curve $\Gamma$. We use this setup to
study the restriction of rank $2$ stable sheaves with prescribed Chern classes
on $X$ to an anticanonical $K3$ surface $S\subset X$. Varying the threefold $X$
containing $S$ gives a rational Lagrangian fibration $$\mathcal{M}_S(2,1,7)
\dashrightarrow \mathbb{P}^3$$ with generic fibre birational to the moduli
space $\mathcal{M}_X(2,1,7)$ of sheaves on $X$. Moreover, we prove that this
rational fibration extends to an actual fibration on a birational model
$\mathcal{M}$ of $\mathcal{M}_S(2,1,7)$.
  In a last part, we use Bridgeland stability conditions to exhibit all
$K$-trivial smooth birational models of $\mathcal{M}_S(2,1,7)$, which consist
in itself and $\mathcal{M}$. We prove that these models are related by a flop,
and we describe the positive, movable and nef cones of $\mathcal{M}_S(2,1,7)$.
