  Most general third-order $3d$ linear gauge vector field theory is considered.
The field equations involve, besides the mass, two dimensionless constant
parameters. The theory admits two-parameter series of conserved tensors with
the canonical energy-momentum being a particular representative of the series.
For a certain range of the model parameters, the series of conserved tensors
include bounded quantities. This makes the dynamics classically stable, though
the canonical energy is unbounded in all the instances. The free third-order
equations are shown to admit constrained multi-Hamiltonian form with the
zero-zero components of conserved tensors playing the roles of corresponding
Hamiltonians. The series of Hamiltonians includes the canonical Ostrogradski's
one, which is unbounded. The Hamiltonian formulations with different
Hamiltonians are not connected by canonical transformations. This means, the
theory admits inequivalent quantizations at the free level. Covariant
interactions are included with spinor fields such that the higher-derivative
dynamics remains stable at interacting level if the bounded conserved quantity
exists in the free theory. In the first-order formalism, the interacting theory
remains Hamiltonian and therefore it admits quantization, though the vertices
are not necessarily Lagrangian in the third-order field equations.
