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We find necessary and sufficient conditions for a local geodesic flow of anaffine connection on a surface to admit a linear first integral. The conditionsare expressed in terms of two scalar invariants of differential orders 3 and 4in the connection. We use this result to find explicit obstructions to theexistence of a Hamiltonian formulation of Dubrovin--Novikov type for a givenone--dimensional system of hydrodynamic type. We give several examplesincluding Zoll connections, and Hamiltonian systems arising fromtwo--dimensional Frobenius manifolds.

First integrals of affine connections and Hamiltonian systems of hydrodynamic type