Here is an example kappa
Out= lambda== -Sqrt[-1-t] || lambda==Sqrt[-1-t]
For different definitions of kappa the form of the result may be very different. Thus providing a single simple method of negating that solution seems risky. Manually I can say that your solution would be
lambda!= -Sqrt[-1-t] && lambda!=Sqrt[-1-t]
is the set of lambda where there are no zeros.
Without knowing more about your kappa I don't see a way to say any more than this.