  The Dirac equation with both scalar and vector couplings describing the
dynamics of a two-dimensional Dirac oscillator in the cosmic string spacetime
is considered. We derive the Dirac-Pauli equation and solve it in the limit of
the spin and the pseudo-spin symmetries. We analyze the presence of cylindrical
symmetric scalar potentials which allows us to provide analytic solutions for
the resultant field equation. By using an appropriate ansatz, we find that the
radial equation is a biconfluent Heun-like differential equation. The solution
of this equation provides us with more than one expression for the energy
eigenvalues of the oscillator. We investigate these energies and find that
there is a quantum condition between them. We study this condition in detail
and find that it requires the fixation of one of the physical parameters
involved in the problem. Expressions for the energy of the oscillator are
obtained for some values of the quantum number $n$. Some particular cases which
lead to known physical systems are also addressed.
