Back in my undergrad I learned that in a dynamical system, if I add a holonomic constraint, I subtract one degree of freedom from the space of configurations. But one can think of situations in which one subtracts more than one: take a single particle, in polar cylindrical coordinates if I fix $r=0$ (one constraint), I have subtracted two degrees of freedom, and in polar spherical coordinates, I have subtracted three degrees of freedom. For values greater than one everything is fine, the holonomic constraint subtracted one degree of freedom.
Is it that for $r=0$ the ones that I didn't fix became sort of "gauge" variables and they count as dynamical variables (because nothing is keeping them from taking whatever value they want), although they do nothing?
How do we know if this is the case for one generalized constraint?