We investigate some aspects of the soliton dynamics in an alpha-helical protein macromolecule within the steric Davydov-Scott model. Our main objective is to elucidate the important role of the helical symmetry in the formation, stability, and dynamical properties of Davydov's solitons in an alpha helix. We show, analytically and numerically, that the corresponding system of nonlinear equations admits several types of stationary soliton solutions and that solitons which preserve helical symmetry are dynamically unstable: once formed, they decay rapidly when they propagate. On the other hand, the soliton which spontaneously breaks the local translational and helical symmetries possesses the lowest energy and is a robust localized entity. We also demonstrate that this soliton is the result of a hybridization of the quasiparticle states from the two lowest degenerate bands and has an inner structure which can be described as a modulated multihump amplitude distribution of excitations on individual spines. The complex and composite structure of the soliton manifests itself distinctly when the soliton is moving and some interspine oscillations take place. Such a soliton structure and the interspine oscillations have previously been observed numerically [A. C. Scott, Phys. Rev. A 26, 578 (1982)]. Here we argue that the solitons studied by Scott are hybrid solitons and that the oscillations arise due to the helical symmetry of the system and result from the motion of the soliton along the alpha helix. The frequency of the interspine oscillations is shown to be proportional to the soliton velocity.