The Mainichi Daily News of 1 June reports that the SDP, having left the governing coalition, may join a no-confidence motion against the government of Yukio Hatoyama. Or then again, they might just abstain, according to the news item. (No chance of passage in any case.)
The SDP also may compete against the governing Democratic Party (DPJ) in the upcoming elections to the second chamber of parliament, although apparently not in all constituencies:
The SDP is expected to further its position as an opposition party, with Fukushima indicating the possibility that the party may back candidates for the House of Councillors election this summer in constituencies where it had heretofore refrained from doing so out of consideration for the Democratic Party of Japan (DPJ), their former coalition partner.
“I’ve heard that we may be fielding candidates in Nagano,” Fukushima said on a television program, regarding a constituency in which DPJ member and Defense Minister Toshimi Kitazawa is expected to run for a fourth term. “I’ve also been hearing about the possibility of somehow backing candidates in Iwate and Kanagawa prefectures. Electoral cooperation (with the DPJ) will only be conducted on a limited basis.” (My emphasis)
So, the SDP has decided to become a sort-of opposition party.