Solutions to a Novel Casimir Equation for the Ito System

We discuss two classes of solutions to a novel Casimir equation associated with the Ito system, a coupled nonlinear wave equation. Both travelling wave solutions and separable self-similar solutions are discussed. In a number of cases, explicit exact solutions are found. Such results, particularly the exact solutions, are useful in that they provide us a baseline of comparison to any numerical simulations. Besides, such solutions provide us a glimpse of the behavior of the Ito system, and hence the behavior of a type of nonlinear wave equation, for certain parameter regimes