Dirac's method of constrained systems applied to a system with a constant of motion

I have learnt that Dirac’s method is used to deal with Hamiltonian systems with constraints between the phase-space variables, for example, $f(q,p)=0$ in a 1-dimensional case, where $f$ is the constraint. Now take the example of a 1D harmonic oscillator. Here we have a constraint of the energy being a constant:
$$
f(q,p,q_0,p_0) = q^2 + p^2 – q_0^2 – p_0^2 =0.
$$
This is a constraint of the form $f(q,p,q_0,p_0)$ and not $f(q,p)$. Textbooks on Dirac’s method deal with the latter type of constraints, i.e. $f(q,p)$. So, does Dirac’s method work on this harmonic oscillator system too?