Consider your favorite continuous time nicely behaved dynamical system. If we linearize at an equilibrium and the largest eigenvalue is negative, then it's stable.
Now let's add to that system the equation $\dot{Z}=0$. Your new system has a conserved quantity ($Z$), and with it comes an eigenvalue of $0$, which means if you increase $Z$, the change doesn't increase or decrease. For your original system, this new equation isn't important to think about. Once your initial condition sets $Z$ the remaining dynamics are what we care about.
I want to write a theorem where I refer to the largest eigenvalue other than these conserved quantity eigenvalues. What should I call them?
For definiteness, consider $\dot{X} = -X + Y$, and $\dot{Y} = \mu X+X^2-Y$, with the additional equation $\dot{Z}=0$. If we study the stability at $0$, a bifurcation will occur as $\mu$ changes, and I want to discuss the eigenvalues not corresponding to the $(0,0,1)$ eigenvector.