  We study the nearest-neighbor distributions of the $k$-body embedded
ensembles of random matrices for $n$ bosons distributed over two-degenerate
single-particle states. This ensemble, as a function of $k$, displays a
transition from harmonic oscillator behavior ($k=1$) to random matrix type
behavior ($k=n$). We show that a large and robust quasi-degeneracy is present
for a wide interval of values of $k$ when the ensemble is time-reversal
invariant. These quasi-degenerate levels are Shnirelman doublets which appear
due to the integrability and time-reversal invariance of the underlying
classical systems. We present results related to the frequency in the spectrum
of these degenerate levels in terms of $k$, and discuss the statistical
properties of the splittings of these doublets.
