= Solution
Work in a gas region with $\rho,p>0$ and physical adiabatic exponent $\gamma>1$. With $\xi=x/t$, each time derivative is $-\xi/t$ times differentiation in $\xi$, and each spatial derivative is $1/t$ times it. The <continuity equation>, <Euler momentum equation> and the <entropy advection equation> for pressure reduce to
$$
(u-\xi)\rho'+\rho u'=0,\qquad (u-\xi)u'+p'/\rho=0,\qquad (u-\xi)p'+\gamma p u'=0.
$$
Their coefficient determinant is $(u-\xi)[(u-\xi)^2-c^2]$, where the <adiabatic sound speed> is $c=\sqrt{\gamma p/\rho}$. If the determinant is nonzero, all derivatives vanish and the solution is uniform.
The apparent third characteristic $u=\xi$ does not give a smooth fan: on such an interval $u'=1$, whereas continuity requires $\rho u'=0$, impossible for $\rho>0$. It cannot support an isolated smooth <entropy> variation either. Subtracting $\gamma$ times the logarithmic density equation from the logarithmic <pressure> equation gives $(u-\xi)(\ln(p/\rho^\gamma))'=0$. A nonzero continuous <entropy> derivative would force $u=\xi$ on an interval and give the same contradiction. Thus the <specific entropy> is constant throughout each smooth branch.
Every nonconstant branch therefore lies in one acoustic characteristic family. Let $\sigma=\pm1$ denote it, with $\xi=u+\sigma c$, so $u-\xi=-\sigma c$. The constant <entropy> means $p=K\rho^\gamma$, $K>0$, and $c'/c=(\gamma-1)\rho'/(2\rho)$. Continuity gives $u'=\sigma c\rho'/\rho=2\sigma c'/(\gamma-1)$. Differentiating $\xi=u+\sigma c$ then proves
$$
\boxed{u'=\frac2{\gamma+1},\qquad c'=\sigma\frac{\gamma-1}{\gamma+1},\qquad s=s_0+c_v\ln K=\text{constant}.}
$$
In particular, $u-2\sigma c/(\gamma-1)=J$ is a constant <Riemann invariant>. All nonconstant smooth local solutions have the explicit form
$$
\boxed{u(\xi)=\frac{2\xi+(\gamma-1)J}{\gamma+1},\quad c(\xi)=\frac{\sigma(\gamma-1)(\xi-J)}{\gamma+1},\quad\rho=\left(\frac{c^2}{\gamma K}\right)^{1/(\gamma-1)},\quad p=K\rho^\gamma,}
$$
on an interval where $c>0$. Along the left-going family ($\sigma=-1$) sound speed decreases with $\xi$; along the right-going family ($\sigma=+1$) it increases. These are the two <self-similar ideal-gas rarefaction> branches. Together with arbitrary uniform positive states they exhaust the smooth local solutions. A global solution can join them piecewise at fan edges, where derivatives generally jump, and can also have discontinuities; those do not constitute additional smooth branches.
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