In this talk we describe a new class of soliton solutions, called umbilic solitons, for certain nonlinear integrable PDE's. 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. 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.
We also introduce a new kind of soliton object for the focusing nonlinear Schrodinger (f)NLS equation which is generated by a collision of two standard optical solitons. One ``sees'' this part of the solution only in the complexification of the soliton Hamiltonian system. It is common to study classes of solutions of the (f)NLS equation by using a complex phase space and real time t and space x variables; in addition to this, we complexify x and t and show that particular complex direction is associated with the new type of soliton object. Even though this object is realized in the complexification of the system, we show that it has real effects. In particular, these effects manifest themselves in the phase shifts of interacting solitons.