If I compute the equilibria of a system of ODEs, is there any significance to a critical point that is undefined for some choice of parameter values?

I'm seeing something weird in a problem I'm working on. As an example of what I mean, say I compute an equilibrium explicitly to get something of the form
$x^* = \frac{\lambda(\beta_1 - 1)}{a(\beta_1-1) + b(\beta_2 - 1)$
for some parameters $\lambda, a, b$. This equilibrium is the only nontrivial equilibrium of the system. When I choose $\beta_1 = 1$, the equilibrium is undefined, but numerically I verify that it's stable, albeit hyperbolic (my system is 2D). What's going on here? Am I just an idiot? I've never had to look for where the equilibria expressions are undefined to find additional steady states in a system