*Are there (known) examples of non-isometric Riemannian metrics on the projective plane that have the same length spectrum?*

This question is related to MO questions http://mathoverflow.net/questions/90530 and http://mathoverflow.net/questions/89882/ . There Zoll surfaces appear as counterexamples of the analogous question on $S^2$ (and spoil all the fun), so maybe one should concentrate on the projective plane where the only Zoll **Riemannian** metric is the the canonical metric. 

**Remark.** Notice that there are tons of very nice reversible Finsler Zoll metrics on the projective plane. Indeed, here is the Busemann recipe to cook up smooth reversible Finsler metrics on $RP^2$ such that all geodesics are projective lines:

1. Take a smooth strictly positive measure on the unit sphere in $\mathbb{R}^3$ that is invariant under the antipodal map.
2. If $x$ and $y$ are distinct, non-antipodal points, let $X$ and $Y$ denote the great circles obtained by intersecting the sphere with the subspaces orthogonal to $x$ and $y$.
3. The union of $X$ and $Y$ cuts the sphere into four connected components. 
4. Define the distance between $x$ and $y$ as the measure of the smallest of these components. 
5. Voilà, you have a metric on the sphere that being invariant under the antipodal map projects down to a metric on the projective plane. It is easy to see that projective lines are geodesics and not too hard to see that it is Finsler.