= Solution
Integrate the initial <Cole-Hopf transformation> and choose a continuous positive normalization:
$$
\psi(\theta,0)=\begin{cases}1,&\theta<0,\\e^{U\theta/(2\epsilon)},&\theta>0.\end{cases}
$$
Split the <heat kernel> convolution at zero, and complete the square in the positive half-line contribution. Define
$$
I_-=\int_\theta^\infty e^{-s^2/(4\epsilon Z)}\,ds,\qquad I_+=\int_{-(\theta+UZ)}^\infty e^{-s^2/(4\epsilon Z)}\,ds,\qquad E=e^{U(\theta+UZ/2)/(2\epsilon)}.
$$
The resulting <heat equation> solution is
$$
\psi=(4\pi\epsilon Z)^{-1/2}(I_-+EI_+).
$$
Differentiating the moving endpoints gives $\partial_\theta I_-=-e^{-\theta^2/(4\epsilon Z)}$ and $E\partial_\theta I_+=e^{-\theta^2/(4\epsilon Z)}$. These terms cancel. Hence $\psi_\theta=(4\pi\epsilon Z)^{-1/2}UEI_+/(2\epsilon)$, and the <viscous Burgers step solution with negative flux> is
$$
\boxed{q=\frac{U}{1+\alpha e^{-U(\theta+UZ/2)/(2\epsilon)}},\qquad \alpha=\frac{I_-}{I_+}.}
$$
The lower limit of $I_-$ is $\theta$, as printed in the original PDF. The converted TeX's $-\theta$ does not give the correct convolution or endpoint cancellation.
For the <vanishing-viscosity limit>, the conservative flux is $F(q)=-q^2/2$ and the <characteristic speed> is $F'(q)=-q$. If $U>0$, the characteristics from the right move left and collide with those on the left. The <Rankine-Hugoniot condition> gives shock speed $[F]/[q]=-U/2$. Thus
$$
q\longrightarrow\begin{cases}0,&\theta<-UZ/2,\\U,&\theta>-UZ/2.\end{cases}
$$
The viscous transition has width $O(\epsilon/U)$, and its centre value is exactly $U/2$ by the <midpoint symmetry of a viscous Burgers step>.
If $U<0$, the right characteristics move right and separate from the left ones. The limit is a <rarefaction wave>,
$$
\boxed{q(\theta,Z)\longrightarrow\begin{cases}0,&\theta\leq0,\\-\theta/Z,&0<\theta<-UZ,\\U,&\theta\geq-UZ.\end{cases}}
$$
This follows either from the <Burgers Riemann problem with negative flux> or the large-argument Gaussian-tail asymptotics of the exact formula. The two signs therefore give a shock and a spreading rarefaction respectively, not two shocks.
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