  We study the SYK$_{2}$ model of $N$ Majorana fermions with random quadratic
interactions through a detailed spectral analysis and by coupling the model to
2- and 4-point sources. In particular, we define the generalized spectral form
factor and level spacing distribution function by generalizing from the
partition function to the generating function. For $N=2$, we obtain an exact
solution of the generalized spectral form factor. It exhibits qualitatively
similar behavior to the higher $N$ case with a source term. The exact solution
helps understand the behavior of the generalized spectral form factor. We
calculate the generalized level spacing distribution function and the mean
value of the adjacent gap ratio defined by the generating function. For the
SYK$_2$ model with a 4-point source term, we find a Gaussian unitary ensemble
behavior in the near-integrable region of the theory, which indicates a
transition to chaos. This behavior is confirmed by the connected part of the
generalized spectral form factor with an unfolded spectrum. The departure from
this Gaussian random matrix behavior as the relative strength of the source
term is increased is consistent with the observation that the 4-point source
term alone, without the SYK$_2$ couplings turned on, exhibits an integrable
spectrum from the spectral form factor and level spacing distribution function
in the large $N$ limit.
