  Long gamma-ray bursts (GRBs) with a plateau phase in their X-ray afterglows
obeys a three-dimensional (3D) relation (Dainotti et al. 2016), between the
rest-frame time at the end of the plateau, Ta, its corresponding X-ray
luminosity, La, and the peak luminosity in the prompt emission, Lpeak, an
extension of the two-dimensional Dainotti relation. This 3D relation identifies
a GRB fundamental plane whose existence we here confirm. We extend the original
analysis with X-ray data untill July 2016 gathering 183 Swift GRBs with
afterglow plateaus and known redshifts. We compare several GRB categories, such
as shorts with extended emission, SEE, X-ray Flashes, GRBs associated with SNe,
a sample of only long-duration GRBs (132), selected from the total sample by
excluding GRBs of the previous categories, and the gold sample, composed by
GRBs with light curves with good data coverage and relatively flat plateaus.
The relation planes for each of these categories are not statistically
different from the gold fundamental plane, with the exception of the SEE, which
are hence identified as a physically distinct class of objects. The gold
fundamental plane has an intrinsic scatter smaller than any plane derived from
the other sample categories. Thus, the distance of any particular GRB category
from this plane becomes a key parameter. Additionally, we tested this 3D
relations by using GRBs observed at high energy, namely the peak luminosity
values derived by the Fermi-Gamma Ray Burst Monitor (GBM). The 3D relation is
also confirmed for GRBs observed by the GBM, thus showing its independence from
the energy range. Furthermore, we computed the several category planes with T*a
as a dependent parameter obtaining for each category smaller intrinsic scatters
(reaching a reduction of $24\%$ for the long GRBs). The fundamental plane is
independent from several prompt and afterglow parameters.
