  To allow for `relativistic'-like core contraction effects, an anisotropic
regularization of steadily-moving straight dislocations of arbitrary
orientation is introduced, with two scale parameters $a_\parallel$ and
$a_\perp$ along the direction of motion and transverse to it, respectively. The
dislocation core shape is an ellipse. When $a_\perp/a_\parallel\to 0$, the
model reduces to the Peierls-Eshelby dislocation, the fields of which are
non-differentiable on the slip plane. For finite $a_\parallel$ and $a_\perp$,
fields are everywhere differentiable. Applying the author's so-called `causal'
Stroh formalism to the model, explicit expressions for the regularized fields
in anisotropic elasticity are derived for any velocity. For faster-than-wave
velocities, Mach-cone angles are found insensitive to the ratio
$a_\parallel/a_\perp$, as must be. However, the larger $a_\parallel$, the
weaker the intensity of the cone branches. An expression is given for the
radiative dissipative force opposed to motion. From this expression, it is
inferred that the concept of a `radiation-free' intersonic velocity can, when
not applicable, be replaced by that of a `least-radiation' velocity.
