The Solicitor General is often involved in cases before the Court agrees to hear them, by filing briefs either supporting or opposing the grant of certiorari. But in none of the Obama eligibility cases has the Solicitor General filed anything, unless you count a two-line "we don't want to file anything" statement (like the one you posted). Filing something like that doesn't require Kagan's recusal, because it doesn't say anything about the merits of the case. ("We waive our right to respond" is not a legal argument that Obama is eligible, and it's not a legal argument that he isn't eligible.)
Had Kagan filed a brief that says the Court shouldn't grant cert., she would be recused, but she hasn't filed such a brief in any of the eligibility cases.
I know you have to look at the particular rules involved, but... I saw a murder case overturned in Texas, because the trial judge, when he was a member of the DA's office, got sent to another court (not the one he usually worked in) one day, to ask that judge to postpone all the cases on the docket, as the prosecutor who worked that court was absent. The Texas courts ruled that any appearance, even one where, so far as we know, the judge/assistant DA never read the docket, and had no idea what cases he was asking to have postponed, was an appearance, and the former assistant DA could not hear the case a judge.
Says Pilsner, who's waiting for heads to explode because he authored a post on a Birther thread that isn't witheringly anti Birther.