  This paper is devoted to the study of the spectral properties of the
Weyl-Dirac or massless Dirac operators, describing the behavior of quantum
quasi-particles in dimension 2 in a homogeneous magnetic field, $B^{\rm ext}$,
perturbed by a chiral-magnetic field, $b^{\rm ind}$, with decay at infinity and
a short-range scalar electric potential, $V$, of the Bessel-Macdonald type.
These operators emerge from the action of a pristine graphene-like QED$_3$
model recently proposed in Eur. Phys. J. B93} (2020) 187. First, we establish
the existence of states in the discrete spectrum of the Weyl-Dirac operators
between the zeroth and the first (degenerate) Landau level assuming that $V=0$.
In sequence, with $V_s \not= 0$, where $V_s$ is an attractive potential
associated with the $s$-wave, which emerges when analyzing the $s$- and
$p$-wave M{\o}ller scattering potentials among the charge carriers in the
pristine graphene-like QED$_3$ model, we provide lower bounds for the sum of
the negative eigenvalues of the operators $|\boldsymbol{\sigma} \cdot
\boldsymbol{p}_{\boldsymbol{A}_\pm}|+ V_s$. Here, $\boldsymbol{\sigma}$ is the
vector of Pauli matrices,
$\boldsymbol{p}_{\boldsymbol{A}_\pm}=\boldsymbol{p}-\boldsymbol{A}_\pm$, with
$\boldsymbol{p}=-i\boldsymbol{\nabla}$ the two-dimensional momentum operator
and $\boldsymbol{A}_\pm$ certain magnetic vector potentials. As a by-product of
this, we have the stability of bipolarons in graphene in the presence of
magnetic fields.
