Third order nonlinear evolution equations, that is the Korteweg-deVries
(KdV), modified Korteweg-deVries (mKdV) equation and other ones are considered:
they all are connected via Baecklund transformations. These links can be
depicted in a wide Baecklund Chart} which further extends the previous one
constructed in [22]. In particular, the Baecklund transformation which links
the mKdV equation to the KdV singularity manifold equation is reconsidered and
the nonlinear equation for the KdV eigenfunction is shown to be linked to all
the equations in the previously constructed Baecklund Chart. That is, such a
Baecklund Chart is expanded to encompass the nonlinear equation for the KdV
eigenfunctions [30], which finds its origin in the early days of the study of
Inverse scattering Transform method, when the Lax pair for the KdV equation was
constructed. The nonlinear equation for the KdV eigenfunctions is proved to
enjoy a nontrivial invariance property. Furthermore, the hereditary recursion
operator it admits [30 is recovered via a different method. Then, the results
are extended to the whole hierarchy of nonlinear evolution equations it
generates. Notably, the established links allow to show that also the nonlinear
equation for the KdV eigenfunction is connected to the Dym equation since both
such equations appear in the same Baecklund chart.Comment: 18 page
