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Most general third-order $3d$ linear gauge vector field theory is considered.The field equations involve, besides the mass, two dimensionless constantparameters. The theory admits two-parameter series of conserved tensors withthe canonical energy-momentum being a particular representative of the series.For a certain range of the model parameters, the series of conserved tensorsinclude bounded quantities. This makes the dynamics classically stable, thoughthe canonical energy is unbounded in all the instances. The free third-orderequations are shown to admit constrained multi-Hamiltonian form with thezero-zero components of conserved tensors playing the roles of correspondingHamiltonians. The series of Hamiltonians includes the canonical Ostrogradski'sone, which is unbounded. The Hamiltonian formulations with differentHamiltonians are not connected by canonical transformations. This means, thetheory admits inequivalent quantizations at the free level. Covariantinteractions are included with spinor fields such that the higher-derivativedynamics remains stable at interacting level if the bounded conserved quantityexists in the free theory. In the first-order formalism, the interacting theoryremains Hamiltonian and therefore it admits quantization, though the verticesare not necessarily Lagrangian in the third-order field equations.

Multi-Hamiltonian formulations and stability of higher-derivative extensions of $3d$ Chern-Simons