  We give a simple geometric interpretation of the mapping of the negative KdV
equation as proposed by Qiao and Li {arXiv:1101.1605 [math-ph], Europhys.
Lett.,94 (2011) 50003} and the Fuchssteiner equation using geometry of
projective connection on S1 or stabilizer set of the Virasoro orbit. We propose
a similar connection between with the higher-order negative KdV equations of
Fuchssteiner type described as and respectively. We study the Painleve and
symmetry analyses of these newly found equations and show that they yield
soliton solutions.
