The Jacobian of the static model is singular in NK SOE

[jpfeifer]

The collinearity comes from a unit root in the model. The bigger problem is the instability. See

Something wrong with the steady state

Yes, there is explosiveness, i.e. one of the stocks (typically capital or bonds) do not have a finite solution. Sometimes that is just a matter of parameters like the elasticities in fiscal rules. At other times, there is a fundamental mistake in the stabilizing mechanisms.