BERLIN — President Joachim Gauck of Germany announced on Monday that he would not seek a second five-year term for the largely ceremonial office, a declaration that could create more problems for Chancellor Angela Merkel.

The announcement is likely to intensify feuding in Ms. Merkel’s so-called grand coalition government of center-right and center-left parties, which is suffering from falling popularity ratings and has been particularly divided on the response to the more than one million migrants who arrived in Germany during the past year.

An opinion poll conducted last week for the Bild newspaper showed that the coalition’s approval rating had fallen below 50 percent for the first time.

Ms. Merkel is expected to seek a fourth term next year, and the survey found that her conservative bloc would command just 30 percent of the votes if an election were held now, and her Social Democrat partners just 19 percent.