We find necessary and sufficient conditions for a local geodesic flow of an affine connection on a surface to admit a linear first integral. The conditions are expressed in terms of two scalar invariants of differential orders 3 and 4 in the connection. We use this result to find explicit obstructions to the existence of a Hamiltonian formulation of Dubrovin–Novikov type for a given one-dimensional system of hydrodynamic type. We give several examples including Zoll connections, and Hamiltonian systems arising from twodimensional Frobenius manifolds.