Pseudoclassical system with gauge and time-reparametrization invariance

We present a pseudoclassical mechanics model which exhibits gauge symmetry and time-reparametrization invariance. As such, first- and second-class constraints restrict the phase space, and the Hamiltonian weakly vanishes. We show that the Dirac conjecture does not hold - the secondary first-class constraint is not a symmetry generator - and only the gauge fixing condition associated with the primary first-class constraint is needed to remove the gauge ambiguities. The gauge fixed theory is equivalent to the Fermi harmonic oscillator extended by a boundary term. We quantize in the deformation quantization and in the Schrödinger representation approaches and observe that the boundary term prepares the system in the state of positive energy.