Lyle Denniston at ScotusBlog has a good analysis of the dispute about the veracity of a Solicitor General's brief to the Court, including this explanation of the underlying case, Nken v. Holder:
Although the government did not win everything it had sought in the Nken case, the Court did rely upon the government’s claim of a policy supposedly favorable to aliens as it turned aside one of the arguments by the non-citizen facing deportation in that case, thus reducing that individual’s chances of avoiding a return to his home country. He feared persecution in his homeland, Cameroon, because his family had been active in political protests against the dictatorial regime in power there.
The individual, Jean Marc Nken, was married to a U.S. citizen, and their son is a citizen. His lawyers took the case to the Supreme Court to get clarification of when a federal appeals court could postpone a deportation order, while the non-citizen involved sought to challenge the legal basis for being sent home. In a 7-2 decision on April 22, 2009, written by Chief Justice John G. Roberts, Jr., the Court accepted Nken’s argument that the traditional factors governing a stay of a court proceeding applied in such a deportation review case, thus turning down the government’s counter argument that deportation could not be postponed unless the order of deportation was more likely than not illegal. In that sense, Nken won.
However, Nken appeared to lose when the Roberts opinion focused upon the question of whether he would face “irreparable injury” if sent home while his court challenge to deportation went forward. The opinion said that “the burden of removal alone” was not enough to satisfy that factor. The opinion then went on to say: “Aliens who are removed [deported] may continue to pursue their petitions for review, and those who prevail can be afforded effective relief by facilitation of their return, along with restoration of the immigration status they had upon removal.” The opinion then cited the Solicitor General’s brief, at page 44.