  We investigate the topological nodal structure of three-dimensional (3D)
ZrTe$_5$ driven by Zeeman splitting as a function of the direction of external
magnetic (B) field by using a Wannier-function-based tight-binding (WFTB) model
obtained from first-principles calculations. It is known that small external
stimuli can drive 3D ZrTe$_5$ into different topological phases including Dirac
semimetal. In order to emphasize the effect of Zeeman splitting, we consider 3D
ZrTe$_5$ in a strong TI phase with a small band gap. With Zeeman splitting
greater than the band gap, the WFTB model suggests that a type-I nodal ring
protected by (glide) mirror symmetry is formed when the B field aligns with the
crystal $a$ or $b$ axes, and that a pair of type-I Weyl nodes are formed
otherwise, when conduction and valence bands touch. We show that a pair of Weyl
nodes can disappear through formation of a nodal ring, rather than requiring
two Weyl nodes with opposite chirality to come together. Interestingly, a
type-II nodal ring appears from crossings of the top two valence bands when the
B field is applied along the $c$ axis. This nodal ring gaps out to form type-II
Weyl nodes when the B field rotates in the $bc$ plane. Comparing the WFTB and
linearized $k \cdot p$ model, we find inadequacy of the latter at some B field
directions. Further, using the WFTB model, we numerically compute the intrinsic
anomalous Hall conductivity ${\sigma_{ac}}$ induced by Berry curvature as a
function of chemical potential and B field direction. We find that
${\sigma_{ac}}$ increases abruptly when the B field is tilted from the $a$ axis
within the $ab$ plane. Our WFTB model also shows significant anomalous Hall
conductivity induced by avoided level crossings even in the absence of Weyl
nodes.
