= Solution
For constant <porosity> and permeability, the equation is $h_T+h_X=\nabla_{X,Y}\cdot(h\nabla_{X,Y}h)$. Under $\tau=T$, $\xi=X-T$, $\eta=Y$, the chain rule gives $h_T=h_\tau-h_\xi$ and $h_X=h_\xi$. Thus the moving-frame <porous medium equation> is
$$
\boxed{h_\tau=\nabla_{\xi,\eta}\cdot(h\nabla_{\xi,\eta}h)
=\tfrac12\nabla_{\xi,\eta}^2(h^2).}
$$
Its conserved mass is $M=\iint h\,d\xi\,d\eta$.
There is a source inconsistency in the printed volume formula. Actual liquid volume is $V=P\iint h\,dx\,dy$. The stated coordinates have <Jacobian determinant>
$$
dx\,dy=\frac{dX\,dY}{\tan^2\theta}
=\frac{d\xi\,d\eta}{\tan^2\theta},
$$
so the <volume Jacobian for downslope stretched coordinates> requires
$$
\boxed{V=P\cot^2\theta\iint h\,d\xi\,d\eta,
\qquad M=\frac{V\tan^2\theta}{P}.}
$$
The printed factor $\tan^2\theta$ in the first expression is its reciprocal and cannot be proved with these coordinates. For example, at $\theta=\pi/6$ an integral equal to one corresponds to actual volume $3P$, whereas the printed formula gives $P/3$. The physical-volume normalization below uses the corrected Jacobian.
Let $r=\sqrt{\xi^2+\eta^2}$. For the source-type axisymmetric <similarity solution>, put
$$
h=\tau^{-a}f(s),\qquad s=r\tau^{-b}.
$$
Mass conservation imposes $a=2b$, and balancing $h_\tau$ against $\nabla\cdot(h\nabla h)$ gives $a+1=2a+2b$. Hence $a=1/2$ and $b=1/4$. The profile equation is
$$
-\tfrac12f-\tfrac14sf'
=\frac1s(sf f')'.
$$
Multiplying by $s$, the left side is $-\tfrac14(s^2f)'$. Integration and regularity at the origin give
$$
sff'=-\tfrac14s^2f.
$$
Where $f>0$, divide by $sf$ to obtain $f'=-s/4$, so
$$
f(s)=C-\frac{s^2}{8},\qquad 0\le s\le\sqrt{8C}.
$$
Set the profile to zero beyond this front. The mass normalization is
$$
M=2\pi\int_0^{\sqrt{8C}}\left(C-\frac{s^2}{8}\right)s\,ds
=4\pi C^2.
$$
Therefore the <two-dimensional Barenblatt profile for quadratic porous-medium diffusion> is
$$
\boxed{h(r,\tau)=\left[
\sqrt{\frac{M}{4\pi\tau}}-\frac{r^2}{8\tau}\right]_+,
\qquad M=\frac{V\tan^2\theta}{P},}
$$
where $[q]_+=\max(q,0)$. Its radius and central thickness are
$$
\boxed{r_N(\tau)=\left(\frac{16M\tau}{\pi}\right)^{1/4},\qquad
h(0,\tau)=\sqrt{\frac{M}{4\pi\tau}}.}
$$
The profile is regular at the center, its flux $-hh_r$ vanishes at the dry edge, and the front speed $r_N'=r_N/(4\tau)$ equals the limiting advective speed $-h_r$ there. Its support shrinks to the release point as $\tau\downarrow0$ while its mass stays $M$, giving the concentrated initial release in the weak sense.
In physical coordinates its center travels at
$$
x_c(t)=\frac{\rho gK\sin\theta}{\mu P}t,
$$
and $r=\tan\theta\sqrt{(x-x_c)^2+y^2}$ with $\tau=\rho gK\sin\theta\tan\theta\,t/(\mu P)$. If the printed volume expression were instead adopted merely as a definition of a formal parameter $\widetilde V$, the same profile would use $M=\widetilde V/(P\tan^2\theta)$; that parameter would equal $V\tan^4\theta$, not the actual released fluid volume.
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