Invariants of P^1 minus a curve

I want to check whether these two objects are isomorphic:
1) P^2 minus the curve x+y+z=0
2) P^2 minus the curve x^2+y^2+z^2=0

The question arose from a statement made that while each of these objects is a P^1, they are not the same - they have different fundamental groups. (Everything is over complex numbers.)
Is there some invariant I can compute in Singular that shows that indeed they are not the same?

Dear Ravi,

unfortunately it seems that we don't have an "out of box" solution for your quite interesting task in Singular right now.

In order to help you we would need some more details, maybe along the following lines:

• how do you do that by hand?
• any standard approaches for that?
• are there any literature references concerning similar problems?

Thanks for trying Singular and
Best regards,
Oleksandr Motsak