Proving Non-Absorbable Constant with Δξgμν

[Morgoth]

Well I have the metric of a sphere given as:
ds²= α (dθ²+ sin²θ dφ² )

And I'm asked to prove that the constant α is non-absorbable by proving there is no solution to this set of equations:
Δ_ξg_μν = ∂g_μν/∂α

where:
Δ_ξg_μν = g_μν;λ ξ^λ + g_λν ξ^λ_;μ + g_μλ ξ^λ_;ν


Finally, I'm asked to provide a second way to prove that α is non-absorbable.



I don't understand what it means by non-absorbable at all. Do you know where am I supposed to find what it is? From books I have Weinberg's "Gravitation and Cosmology Principles and appls of GTR". In addition, any hint would be appreciated...
Thank you..

 

[Morgoth]

This is my 1st try to solve it, the equations I ended up are the 5 bellow, which cannot be solved (i guess). The last one reads ξ²_,2=1/2α..

I also got a problem with 0=1...

What do you think?