(Updates at 0530 GMT)

By Ankur Banerjee

SINGAPORE, July 1 (Reuters) - Uncertainty over the U.S.

rates outlook kept Asian shares steady on Monday, while the euro

rose after the first-round voting in France's shock snap

election was won by the far-right, albeit with a smaller share

than some polls had projected.

The shock vote has unsettled markets as the far-right, as

well as the left-wing alliance that came second on Sunday, have

pledged big spending increases at a time when France's high

budget deficit has prompted the EU to recommend disciplinary

steps.

On Monday, the euro was 0.41% higher, while

European stock futures rose 1.4% and French OAT bond

futures gained 0.15% as investors digested the better

than feared results, although uncertainty remained.

"There is a sense of relief that the first round of the

French elections weren't as comprehensively in Le Pen's favour

as the polls indicated," said Tony Sycamore, market analyst at

IG.

"This raises hopes that National Rally won't win an outright

majority, nor be in a position to open the purse strings, a

proposition which had the French bond market and the euro

looking nervously over their shoulders."

Exit polls showed Marine Le Pen's National Rally (RN)

winning around 34% of the vote, comfortably ahead of leftist and

centrist rivals but the chances of eurosceptic, anti-immigrant

RN winning power next week will depend on the political

dealmaking by its rivals over the coming days.

The focus now shifts to next Sunday's runoff and will depend

on how parties decide to join forces in each of France's 577

constituencies for the second round, and could still result in a

majority for RN.

"Investors are concerned that if the far-right National

Rally party wins a majority, this could set the stage for France

to clash with the EU, which could disrupt Europe's markets and

the euro sharply," said Vasu Menon, managing director of

investment strategy at OCBC.

In Asia, the MSCI's broadest index of Asia-Pacific shares

outside Japan was 0.04% lower, in a subdued

start to the second half of the year having risen 7% so far in

2024.

China stocks were mixed, with blue-stocks down

0.19% and the Shanghai Composite index up 0.31% after a

mixed set of data.

Factory activity among smaller Chinese manufacturers grew at

the fastest pace since 2021 thanks to overseas orders, while

weak domestic demand and trade frictions had led to another

industrial sector contraction.

On the macro side, the spotlight remains on if and when the

Federal Reserve will start cutting rates in the wake of data on

Friday showing U.S. monthly inflation was unchanged in May.

In the 12 months through May, the PCE price index increased

2.6% after advancing 2.7% in April. Last month's inflation

readings were in line with economists' expectations but they

remain above the Fed's 2% target for inflation.

Still, markets are clinging to expectations of at least two

rate cuts from the Fed this year with a cut in September pegged

in at 63% probability, CME FedWatch tool showed.

Investor focus this week will be on the minutes of the Fed's

June meeting that will offer more clues on the central bank's

thinking before the spotlight switches to payrolls data on

Friday. The Feb in June projected just one rate cut in 2024.

Among currencies, the yen traded slightly weaker

at 161.06 per dollar after skidding to 161.27 on Friday, its

weakest level since late 1986, keeping traders on edge for signs

of intervention from the Japanese authorities.

A quarterly central bank survey showed on Monday the

business mood in Japan's service sector soured in June, while a

rare unscheduled downgrade to the country's GDP data also showed

the economy shrank more than reported in the first quarter.

The euro touched a more than two week high of $1.076175 in

early Asian hours, pushing the dollar index, which

measures the U.S. unit against six rivals, a touch lower at

105.61.

In commodities, oil prices edged higher, with Brent futures

0.39% higher at $85.33 per barrel and U.S. West Texas

Intermediate crude futures up 0.39% at $81.86.

(Editing by Stephen Coates and Miral Fahmy)