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NEW YORK—Wall Street began the third quarter with an erratic session and modest gain Tuesday after a mix of news made it clear the country is still deep in economic problems but may have some positive trends—including some better than expected sales for General Motors Corp.

Prices rose early in the session, then turned sharply lower for much of the day and then recovered in late afternoon. The uneven performance wasn’t surprising—some bargain hunting was to be expected after a dismal first half, and in particular, a dismal June.

The session brought more discouraging new...