The Dirac-Bergmann algorithm effectively isolates the physical degrees of freedom of a system, by changing from Poisson brackets $\{\cdot,\cdot\}_\mathrm{PB}$ to Dirac brackets $\{\cdot,\cdot\}_\mathrm{D}$.
Quick overview: let $\chi_i\approx0$ be the constraints. We demand that $\chi_i=\chi_i(q,p)$ and write $M_{ij}\equiv \{\chi_i,\chi_j\}_\mathrm{PB}$ for every second class constraint. Finally, the Dirac brackets are given by $\{\cdot,\cdot\}_\mathrm{D}=\{\cdot,\cdot\}_\mathrm{PB}-\{\cdot,\chi_i\}M^{ij}\{\chi_j,\cdot\}$.
My question: if we have a velocity-dependent constraint, $\chi_0=\chi_0(q,\dot q)$, and $p=p(\dot q)$ is not invertible (singular Legendre transform), then the Poisson bracket $\{\chi_0,\cdot\}_\mathrm{PB}$ is not defined. Does this mean it is impossible to define $\{\cdot,\cdot\}_\mathrm{D}$?