DONETSK, Ukraine—In a city filled to the brim with pro-Russia insurgents bracing for a showdown with the Ukrainian military, public transport continues to run, state pensions are paid, parks are impeccably tended and the municipal water supply is being restored.

Much of that is because Donetsk, despite hosting a separatist movement hostile to all things Ukrainian, still has its popular, elected mayor, who has kept his job by refusing to take sides.

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