WRAPUP 10-Ukraine says it downs Russian missiles, UN pushes for nuclear plant's safety

(Adds Zelenskiy on Russian missiles, U.N. chief proposals for Zaporozhzhia nuclear plant safety)

* Ukraine waging counter-offensive in south against Russians

* U.N. seeks security zone near damaged Zaporozhzhia nuclear plant

By Pavel Polityuk and Francois Murphy

KYIV/VIENNA, Sept 6 (Reuters) - Ukrainian forces shot down five Russian cruise missiles on Tuesday, most of them in the south, President Volodymyr Zelenskiy said, but he made no mention of a new military success in the east hinted at by officials earlier.

"This morning alone, five of six Russian X-101 missiles were downed," Zelenskiy said in an evening address. "This is a costly loss for Russia, and it saves many Ukrainian lives. Four of these missiles were downed by the "South" district of the air command."

Reuters was not able to independently verify Zelenskiy's statement and there was no immediate response from Russia.

Though Ukrainian officials did not give specific details, several posts on social media from military bloggers and witnesses reported fighting around Balakliia, an eastern town of 27,000 people that lies between Kharkiv and Russian-held Izyum, a city with a major railway hub used by Moscow to supply its forces. One tweet by a Zelenskiy adviser spoke of "great news" coming from the president on the operation in Kharkiv region.

Reuters was unable to independently verify the reports posted on social media.

Little information has emerged about progress of the main Ukrainian offensive in the southern Kherson region, with Kyiv barring journalists from the front line and releasing only limited reports to preserve the element of surprise.

Russia says it has repelled the Kherson assault, but Ukraine has reported steady success.

Western military experts say Ukraine's aim in the south appears to be to trap thousands of Russian troops on the west bank of the wide Dnipro River and cut them off by destroying their rear supply lines.

The announcement of a simultaneous Ukrainian advance near Kharkiv was an indication that Russian troops were having difficulty reinforcing along the front, said Mark Hertling, a retired former U.S. commander of ground forces in Europe.

"This confirms RU inability to maneuver forces between theater locations & within their force's defensive 'stance' to counter Ukrainian offensive actions," Hertling tweeted.

There have also been increasing attacks by Ukrainians in Russian-occupied areas against Moscow-installed authorities.

On Tuesday, the Russian-installed commandant of the southern port of Berdiansk was badly injured when his car was blown up outside the city administration building, pro-Russian local officials said, blaming Ukraine for the attack.