  In [22] Milnor proved that a real analytic map $f\colon (R^n,0) \to (R^p,0)$,
where $n \geq p$, with an isolated critical point at the origin has a fibration
on the tube $f|\colon B_\epsilon^n \cap f^{-1}(S_\delta^{p-1}) \to
S_\delta^{p-1}$. Constructing a vector field such that, (1) it is transverse to
the spheres, and (2) it is transverse to the tubes, he "inflates" the tube to
the sphere, to get a fibration $\varphi\colon S_\epsilon^{n-1} \setminus
f^{-1}(0) \to S^{p-1}$, but the projection is not necessarily given by $f/
\|f\|$ as in the complex case.
  In the case $f$ has isolated critical value, in [9] it was proved that if the
fibres inside a small tube are transverse to the sphere $S_\epsilon$, then it
has a fibration on the tube. Also in [9], the concept of $d$-regularity was
defined, it turns out that $f$ is $d$-regular if and only if the map
$f/|f\|\colon S_\epsilon^{n-1} \setminus f^{-1}(0) \to S^{p-1}$ is a fibre
bundle equivalent to the one on the tube.
  In this article, we prove the corresponding facts in a more general setting:
if a locally surjective map $f$ has a linear discriminant $\Delta$ and a
fibration on the tube $f|\colon B_\epsilon^n \cap f^{-1}(S_\delta^{p-1}
\setminus \Delta) \to S_\delta^{p-1} \setminus \Delta$, then $f$ is $d$-regular
if and only if the map $f/ \|f\|\colon S_\epsilon^{n-1} \setminus
f^{-1}(\Delta) \to S^{p-1} \setminus \mathcal{A}$ (with $\mathcal{A}$ the
radial projection of $\Delta$ on $S^{p-1}$) is a fibre bundle equivalent to the
one on the tube. We do this by constructing a vector field $\tilde{w}$ which
inflates the tube to the sphere in a controlled way, it satisfies properties
analogous to the vector field constructed by Milnor in the complex setting:
besides satisfying (1) and (2) above, it also satisfies that $f/ \|f\|$ is
constant on the integral curves of $\tilde{w}$.
