President Gurbanguly Berdymukhamedov's 40-year-old son Serdar, a deputy prime minister, is seen as a likely successor.
Voting goes underway in a parliamentary election which could indicate a path towards hereditary succession in the gas-rich former Soviet state.
Kurbanguly Berdymukhamedov, 59, has run the former Soviet republic of five million people for a decade amid slow economic growth and falling exports.
Voting is underway in a one-sided election in which the incumbent Kurbanguly Berdymukhamedov is facing eight other candidates.
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