In this paper we describe a new class of soliton solutions, called umbilic solitons, for certain nonlinear integrable PDES. These umbilic solitons have the property that as the space variable x tends to infinity, the solution tends to a periodic wave, and as x tends to minus infinity, it tends to a phase shifted wave of the same shape. The equations admitting solutions in this new class include the Dym equation and equations in its hierarchy. The methods used to find and analyse these solutions are those of algebraic and complex geometry. We look for classes of solutions by constructing associated finite-dimensional integrable Hamiltonian systems on Riemann surfaces. In particular, in this setting we use geodesics on n-dimensional quadrics to find the spatial, or x-flow, which, together with the commuting t-flow given by the equation itself, defines new classes of solutions. Amongst these geodesics, particularly interesting ones are the umbilic geodesics, which then generate the class of umbilic soliton solutions. This same setting also enables us to introduce another class of solutions of Dym-like equations, which are related to elliptic and umbilic billiards.