* Grybauskaite broadly backs existing austerity measures

* President can influence economic policy, veto budget laws

* She says some tax rises a mistake, wants to push exports

By Nerijus Adomaitis

VILNIUS, May 18 (Reuters) - European Union budget commissioner Dalia Grybauskaite on Sunday became Lithuania's first woman president, vowing to help the recession-hit Baltic nation fight an economic crisis.

The 53-year-old, a tough-talking former Lithuanian finance minister, ran as an independent candidate, enhancing her popularity in contrast to the main political parties, whose standing has been hit by the economic downturn and allegations of corruption.

The country, an EU and NATO member since 2004, experienced its worst post-Soviet rioting in January.

In Lithuania, the president is the head of state and formally appoints the prime minister and the cabinet.