  We present a multi-parameter family of rational solutions to the complex
Korteweg-de Vries(KdV) equations. This family of solutions includes particular
cases with high-amplitude peaks at the centre, as well as a multitude of cases
in which high-order rogue waves are partially split into lower-order
fundamental components. We present an empirically-found symmetry which
introduces a parameter controlling the splitting of the rogue wave components
into multi-peak solutions, and allows for nonsingular solutions at higher order
in certain cases.
