BRUSSELS — The leaders of 16 countries survived the “frank talk” of an emergency meeting on immigration and emerged with a veneer of common purpose Sunday as the European Union enters a potentially vital week for its unity.

German Chancellor Angela Merkel said the meeting produced “a lot of goodwill” to resolve differences over how to manage the refugees and economic migrants who are willing to risk their lives to reach Europe.

The leaders of several more countries joined Italy and France in endorsing the idea of setting up centers in Africa to screen potential asylum-seekers for eligibility before they set out for another continent.

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