  We investigate the possibility to discriminate between different pairs of CP
non-conserving mechanisms inducing the neutrinoless double beta
$\betabeta$-decay by using data on $\betabeta$-decay half-lives of nuclei with
largely different nuclear matrix elements (NMEs). The mechanisms studied are:
light Majorana neutrino exchange, heavy left-handed (LH) and heavy right-handed
(RH) Majorana neutrino exchanges, lepton charge non-conserving couplings in
SUSY theories with R-parity breaking giving rise to the "dominant gluino
exchange" and the "squark-neutrino" mechanisms. The nuclei considered are
$^{76}$Ge, $^{82}$Se, $^{100}$Mo, $^{130}$Te and $^{136}$Xe. Four sets of
nuclear matrix elements (NMEs) of the decays of these five nuclei, derived
within the Self-consistent Renormalized Quasiparticle Random Phase
Approximation (SRQRPA), were employed in our analysis. While for each of the
five single mechanisms discussed, the NMEs for $^{76}$Ge, $^{82}$Se, $^{100}$Mo
and $^{130}$Te differ relatively little, the relative difference between the
NMEs of any two nuclei not exceeding 10%, the NMEs for $^{136}Xe$ differ
significantly from those of $^{76}$Ge, $^{82}Se$, $^{100}$Mo and $^{130}$Te,
being by a factor $\sim (1.3 - 2.5)$ smaller. This allows, in principle, to
draw conclusions about the pair of non-interfering (interfering) mechanisms
possibly inducing the $\betabeta$-decay from data on the half-lives of
$^{136}Xe$ and of at least one (two) more isotope(s) which can be, e.g., any of
the four, $^{76}Ge$, $^{82}Se$, $^{100}Mo$ and $^{130}Te$. Depending on the
sets of mechanisms considered, the conclusion can be independent of, or can
depend on, the NMEs used in the analysis. The implications of the EXO lower
bound on the half-life of $^{136}Xe$ for the problem studied are also
exploited.
