  Bose-Einstein condensates (BECs) confined in a two-dimensional (2D) harmonic
trap are known to possess a hidden 2D Schr\"odinger symmetry, that is, the
Schr\"odinger symmetry modified by a trapping potential. Spontaneous breaking
of this symmetry gives rise to a breathing motion of the BEC, whose oscillation
frequency is robustly determined by the strength of the harmonic trap. In this
paper, we demonstrate that the concept of the 2D Schr\"odinger symmetry can be
applied to predict the nature of three dimensional (3D) collective modes
propagating along a condensate confined in an elongated trap. We find three
kinds of collective modes whose existence is robustly ensured by the
Schr\"odinger symmetry, which are physically interpreted as one breather mode
and two Kelvin-ripple complex modes, i.e., composite modes in which the vortex
core and the condensate surface oscillate interactively. We provide analytical
expressions for the dispersion relations (energy-momentum relation) of these
modes using the Bogoliubov theory [D. A. Takahashi and M. Nitta, Ann. Phys.
354, 101 (2015)]. Furthermore, we point out that these modes can be interpreted
as "quasi-massive-Nambu-Goldstone (NG) modes", that is, they have the
properties of both quasi-NG and massive NG modes: quasi-NG modes appear when a
symmetry of a part of a Lagrangian, which is not a symmetry of full a
Lagrangian, is spontaneously broken, while massive NG modes appear when a
modified symmetry is spontaneously broken.
