  We study the level statistics of a non-integrable one dimensional interacting
fermionic system characterized by the GOE distribution. We calculate
numerically on a finite size system the level spacing distribution $P(s)$ and
the Dyson-Mehta $\Delta_3$ correlation. We observe that its low energy spectrum
follows rather the Poisson distribution, characteristic of an integrable
system, consistent with the fact that the low energy excitations of this system
are described by the Luttinger model. We propose this Random Matrix Theory
analysis as a probe for the existence and integrability of low energy effective
Hamiltonians for strongly correlated systems.
