  The Hamiltonian dynamics of spherically symmetric massive thin shells in the
general relativity is studied. Two different constraint dynamical systems
representing this dynamics have been described recently; the relation of these
two systems is investigated. The symmetry groups of both systems are found. New
variables are used, which among other things simplify the complicated system a
great deal. The systems are reduced to presymplectic manifolds Gamma_1 and
Gamma_2, lest non-physical aspects like gauge fixings or embeddings in extended
phase spaces complicate the line of reasoning. The following facts are shown.
Gamma_1 is three- and Gamma_2 is five-dimensional; the description of the shell
dynamics by Gamma_1 is incomplete so that some measurable properties of the
shell cannot be predicted. Gamma_1 is locally equivalent to a subsystem of
Gamma_2 and the corresponding local morphisms are not unique, due to the large
symmetry group of Gamma_2. Some consequences for the recent extensions of the
quantum shell dynamics through the singularity are discussed.
