By John O'Donnell

FRANKFURT (Reuters) - Attacks on women in Cologne and other German cities on New Year's Eve have prompted more than 600 criminal complaints, with police suspicion resting on asylum seekers, putting pressure on Chancellor Angela Merkel and her open door migrant policy.

The attacks, mostly targeting women and ranging from theft to sexual molestation, have prompted a highly-charged debate in Germany about its welcoming stance for refugees and migrants, more than one million of whom arrived last year.

The sudden nature of the violent attacks and the fact that they stretched from Hamburg to Frankfurt prompted Germany's justice minister Heiko Maas to speculate in a newspaper that they had been planned or coordinated.

The debate on migration will be further fueled by the acknowledgement by the authorities in North Rhine-Westphalia that a man shot dead as he tried to enter a Paris police station last week was an asylum seeker with seven identities who lived in Germany.

In Cologne, police said on Sunday that 516 criminal complaints had been filed by individuals or groups in relation to assaults on New Year's Eve, while police in Hamburg said 133 similar charges had been lodged with the north German city.

Frankfurt also registered complaints, although far fewer.

The investigation in Cologne is focused largely on asylum seekers or illegal migrants from north Africa, police said. They arrested one 19-year-old Moroccan man on Saturday evening.

In Cologne, where a 100-strong force of officers continued their investigations, around 40 percent of the complaints included sexual offences, including two rapes.

DWINDLING TRUST

The attacks, which prompted violent far-right protests on Saturday, threatens to further erode confidence in Merkel, and could stoke support for the anti-immigrant Alternative for Germany (AfD) party ahead of three key state elections in March.

Merkel's popularity has dwindled as she refused to place a limit on the influx of refugees.

A survey sponsored by state broadcaster ARD showed that while 75 percent of those asked were very happy with Merkel's work in April last year, only 58 percent were pleased now.

Almost three quarters of those polled said migration was the most important issue for the government to deal with in 2016.

The Cologne attacks also heated up the debate on immigration in neighboring Austria.

"What happened in Cologne is unbelievable and unacceptable," Austrian Interior Minister Johanna Mikl-Leitner, a member of the conservative People's Party that is junior coalition partner to the Social Democrats, told newspaper Oesterreich.

There had been a handful of similar incidents in the border city of Salzburg. "Such offenders should be deported," she said, backing a similar suggestion by Merkel.

Swiss media contained numerous stories about sexual assaults on women by foreigners, fuelling tensions ahead of a referendum next month that would trigger the automatic deportation of foreigners convicted of some crimes.

In Germany, on Monday, a regional parliamentary commission will quiz police and others about the events on New Year's Eve in Cologne.

The anti-Islam PEGIDA, whose supporters threw bottles and fire crackers at a march in Cologne on Saturday before being dispersed by riot police, will later hold a rally in the eastern German city of Leipzig.

The far-right will likely seize on reports that the Paris attacker, who was shot last week as he wielded a meat cleaver and shouted "Allahu akbar" (God is Greatest), was known to police for drug dealing and harassing women.

He had an apartment in an accommodation center for asylum seekers in Recklinghausen, north of Cologne, where he had painted the symbol of Islamic state on the wall of two rooms.

(Additional reporting by Ralf Bode in Berlin and Michael Shields in Zurich; Editing by Ros Russell)