  We study the SYK model with an extra constant source, \.i.e. a constant
matrix or equivalently a diagonal matrix with only one non-zero entry
$\lambda_1$. By using methods from analytic combinatorics, we find exact
expressions for the moments of this model. We further prove that the spectrum
of this model can have a gap when $\lambda_1>\lambda^c_1$, thus exhibiting a
phase transition in $\lambda_1$. In this case, a single isolated eigenvalue
splits off from SYK's eigenvalues distribution. We located this single
eigenvalue by analyzing the singular behavior of a supercritical functional
composition scheme. In certain limit our results recover the ones of random
matrices with non-zero mean entries.
