In this work we show that on the space of solutions of a certain class of fourth-order ODEs, u = (s, u, u , u , u ), a four-dimensional conformal metric, g ab , can be constructed such that the four-dimensional eikonal equation, g ab u ,a u ,b = 0, holds. Furthermore, we remark that this structure is invariant under contact transformations. Our general results are applied to the Minkowski spacetime.