We study lengths of helices of orders 3 and 4 which are generated by some Killing vector fields on a complex projective plane and on a complex hyperbolic plane. We consider the moduli space of such helices under the congruence relation and give a lamination structure on this space which are closely related with the length spectrum. This shows that the moduli space does not form a canonical building structure with respect to the length spectrum.
Toshiaki ADACHI. "Essential Killing helices of order less than five on a non-flat complex space form." J. Math. Soc. Japan 64 (1) 1 - 21, January, 2012. https://doi.org/10.2969/jmsj/06410001