Geometrical and gauge equivalence of the generalized Hirota, Heisenberg and Wkis equations with linear inhomogeneities

Integrable evolution equations can take several equivalent forms in a geometrical sense. Here we consider the equivalence of generalized versions involving linear inhomogeneities of three important nonlinear evolution equations, namely the Hirota, Heisenberg ferromagnetic spin and Wadati-Konno-Ichikawa-Shimizu (WKIS) equation through a moving helical space curve formalism and stereographic representation. From the geometrical consideration, we also construct suitable (2 × 2)-matrix linear eigenvalue equations, involving however non-isospectral flow: the eigenvalues evolve in time. However, these systems are also gauge equivalent. We briefly analyse the scattering problem and show that infinite number of constants of motion can exist for these systems.