The European Central Bank's injection of half a trillion euros into banks has improved market conditions for lenders to bolster capital buffers, the EU's banking watchdog said on Wednesday.

It's important that the banks, supported by these unlimited funds, improve their capital. These two things are two sides of the same coin, European Banking Authority (EBA) Chairman Andrea Enria told a hearing on regulation in the French Senate.

The ECB is set to offer more unlimited, 3-year loans on February 29, helping to ease market worries over funding for now. The EBA has told 31 banks in the EU to lift their core capital ratio to at least 9 percent by the end of June.

The banks have taken our recommendation seriously. The process of recapitalisation is going well, Enria said.

French bank BNP Paribas , on the EBA list, said on Wednesday it has already complied, helping to send its shares higher.

Enria was, however, worried about banks no longer lending to companies outside their national market. There is a process of renationalisation, a repatriation of credit and that brings risks. We could see the single market fragmented more than before, he said.

Francois Perol, chief executive of French bank BPCE, confirmed the trend. There is a preference for customers closer to home, he said.

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Enria defended EBA's mandate to introduce a single rulebook for all banks across the 27-country EU, a move which worries countries like Britain and Sweden which want freedom to continue applying tougher capital standards locally.

Many authorities have some doubts over the concept of the single rulebook in Europe, Enria said.

But EBA found that applying four different methods used across the EU for calculating core capital ratios for the same bank arrived a wide range of ratios, between 7 and 10 percent.

We must have the same rules, but with some flexibility, Enria said.

Perol said markets were already forcing lenders to comply now with the tougher Basel III bank capital rules that were not due to take full effect until 2019.

To meet such market expectations, banks must be allowed to widen beyond government bonds and cash the assets Basel III allows in mandatory liquidity buffers, Perol said.

In a veiled swipe at Britain, Daniele Nouy, secretary general of French bank supervisor ACP, also backed a single rulebook.

Nouy said newly virtuous countries may want to have flexibility to impose tougher rules now, but in time could use such leeway to offer local lenders competitive advantages.

The EU bank capital rules already give national supervisors room for discretion, she added.

(Writing by Huw Jones; Editing by Helen Massy-Beresford)