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The pair disagreed on China, Iran, Iraq and the future of Nato, marring efforts by US and European leaders to declare that transatlantic relations had entered a new era of harmony.
As Mr Bush attended consecutive summits with Nato and EU heads of government Brussels police fired water cannon to break up demonstrations against his presence.
In Moscow, President Putin took issue with Mr Bush’s criticism of his drift towards authoritarianism, saying that Russia would develop its own form of democracy without foreign interference.
In Brussels, the clearest disagreement was over the EU’s plans to lift its arms embargo against China. “With regard to China, Europe intends to remove the last obstacles to its relations with this important country,” M. Chirac insisted.
But Mr Bush spoke of his “deep concern” that a transfer of military technology would change the balance of relations between China and Taiwan, a country US troops might one day have to protect.
European leaders believe they can meet American concerns with a strict code of conduct on what could be exported, and allowing officials to check dual use items were not being used in illicit weaponry.
But Mr Bush expressed no enthusiasm for such a system, and said he would not intervene to prevent the US Congress retaliating by blocking US military sales to Europe.
“The Congress will be making the decisions as to how to react to what will be perceived by some as a technology transfer to China,” he said.
On Iran, M.Chirac urged Mr Bush to back European efforts to dissuade Tehran from pursuing nuclear weapons by granting it economic and trade benefits. M Chirac said that Iran should be allowed to buy civil aircraft engines and to join the World Trade Organisation. “I don’t see why that should not be done and I said so to the President of the United States,” he said.
More specifically, Gerhard Schröder, the German Chancellor, said that the EU wanted to sell European Airbuses to Iran.
But Mr Bush is opposed to Britain, France and Germany, the three EU countries negotiating directly with Tehran, offering such incentives.
Mr Bush declared: “We liberated Iraq. And that decision has been made, it’s over with and now it is time to unify for the sake of peace. The key now is to put that behind us and to focus on helping the new democracy succeed.”
All 26 Nato members pledged contributions to an alliance training mission in Baghdad. The EU and US agreed to hold a joint conference to boost international efforts to rebuild Iraq. “We all agreed to bury the hatchet,” said a senior US Administration official.
But Mr Bush suggested he was still irked by Germany, France and Spain still refusing to help train Iraqi security forces inside Iraq.
Mr Bush and M Chirac also appeared to differ on the future of Nato, an issue raised earlier this month by Herr Schröder who said that the trans-Atlantic alliance needed a new forum to discuss political and security issues.
M Chirac said that he agreed with Herr Schröder, the EU should increasingly become the forum for Europe and the US to discuss their security. He said: “Europe is steadily building up its defence capability. This development is a positive opportunity for our alliance, for a stronger, more united Europe plainly implies a stronger, more effective Atlantic alliance.”
Mr Bush will today have lunch with Herr Schröder in Germany, where hundreds demonstrated against his presence yesterday. Tomorrow, he meets Mr Putin in Slovakia. Mr Bush said he would relay the concerns about Russia that he heard from Baltic leaders.
But Mr Putin told Slovak media that Russia would pursue its own brand of democracy. “The fundamental principles and institutions of democracy must be adapted to the realities of Russian life today, to our traditions and our history,” he said. “And we will do this ourselves.”
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