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GLOBAL MARKETS-Asian shares flat, still on track for winning week

* Dollar steady, with DXY poised for gaining week

* U.S. crude edges up, pulls further away from this week's lows

TOKYO, June 23 (Reuters) - Asian shares balked at the starting gate on Friday but remained on track for a weekly gain, while crude oil prices pulled away from this week's 10-month lows.

MSCI's broadest index of Asia-Pacific shares outside Japan was flat in early trade, and up 0.4 percent for the week.

Japan's Nikkei stock index edged down slightly, but was on track to log a rise of 0.8 percent in a week in which it touched its highest levels since August 2015.

The dollar index, which tracks the greenback against a basket of six major rivals, was down 0.1 percent at 97.546 , but up 0.4 percent for the week.

The euro was down slightly on the day at $1.1148, and down 0.4 percent for the week, while the dollar was steady against the yen at 111.28, up 0.4 percent for the week.

"We're getting close to the end of the month, and fundamentals aside, there will be people selling dollars, so it will be easy for the yen to strengthen next week," said Mitsuo Imaizumi, Tokyo-based chief foreign exchange strategist for Daiwa Securities.

"We also need to keep an eye on the healthcare debate in Washington, because political turmoil tends to undermine the dollar," he said.

U.S. Senate Republicans offered a bill on Thursday to overhaul Obamacare, the next phase in the party's long war against the 2010 law enacted by then-President Barack Obama, though it remained unclear if the bill has enough support to pass the Senate.

On Wall Street overnight, U.S. shares put in a mixed performance, though the S&P healthcare index rose 1 percent and hit its fifth consecutive record close following the release of Senate Republicans' bill.

U.S. economic data on Thursday showed the number of Americans filing for unemployment benefits rose slightly last week, but remained at levels consistent with a tight labor market. Home prices also increased in April more than expected.

The Mexican peso added 0.1 percent after soaring 1 percent on Thursday as Mexico's central bank board raised interest rates, saying it wanted to anchor inflation expectations and take into account last week's move by the U.S. Federal Reserve to hike borrowing costs.

Crude oil futures pulled further away from this week's lows, though market sentiment remained fragile amid a global crude glut that has persisted despite OPEC-led output cuts.

Brent crude was up 0.3 percent at $45.36 a barrel. U.S. crude futures also rose 0.3 percent to $42.86 a barrel.

(Reporting by Tokyo markets team; Editing by Eric Meijer)