Vladimir Matveev - "Projective vector fields: Solution of Lie and Schouten problems".

In 1882 Sophus Lie asked to describe all 2-dimensional metrics admitting vector fields whose flows sends 
(unparameterized) geodesics to geodesics. I will explain the solution of this problem (part of the results are joint 
with R.L. Bryant and G. Manno). The results are published in arXiv:0705.3592, arXiv:0802.2344, and arXiv:0802.2346. 
The solution uses a description of metrics whose geodesic flows admit integrals quadratic in velocities; 
I will give an overview of this subject.

Besides this theory, the solution consists of a trick and huge calculations. I will explain the trick and hope you will explain me 
what other classical problems one can hope to solve by using similar calculations.