  We study the existence and nonexistence of singular solutions to the equation
$u_t-\Delta u - \frac{\kappa}{|x|^2}u+|x|^\alpha u|u|^{p-1}=0$, $p>1$, in
$\R^N\times[0,\infty)$, $N\ge 3$, with a singularity at the point $(0,0)$, that
is, nonnegative solutions satisfying $u(x,0)=0$ for $x\ne0$, assuming that
$\a>-2$ and $\kappa<\left(\frac{N-2}2\right)^2$. The problem is transferred to
the one for a weighted Laplace-Beltrami operator with a non-linear absorbtion,
absorbing the Hardy potential in the weight. A classification of a singular
solution to the weighted problem either as a {\it source solution} with a
multiple of the Dirac mass as initial datum, or as a unique {\it very singular
solution}, leads to a complete classification of singular solutions to the
original problem, which exist if and only if
$p<1+\frac{2(2+\alpha)}{N+2+\sqrt{(N-2)^2-4\kappa}}$.
