The immigration system lauded by Boris Johnson, Liam Fox and other top Brexit backers has been rubbished by Theresa May in a move that could spark fresh Cabinet divisions.

The Prime Minister said there is “no silver bullet” for tackling immigration and that it was not clear that a proposal for an Australian-style points based system, that formed the corner-stone of the Leave campaign’s referendum pitch, actually worked.

She also paved the way for allowing EU citizens preferential treatment in any Brexit deal, and would not commit to other key promises of the leave campaign - such as removing VAT on fuel or giving freed-up funds to the NHS.

Mrs May also refused to rule out paying money into the EU budget, as other non-member countries that seek access to the single market do.

The Australian style points-based immigration system became the flagship policy of the Leave campaign during the referendum, touted as the answer to the UK’s immigration problems.

Mr Johnson, now Mrs May’s Foreign Secretary, regularly expressed support as he travelled the UK on the Brexit campaign bus. International Trade Secretary Mr Fox made the system one of the first policies of his Tory leadership bid.

International Development Secretary Priti Patel, Transport Secretary Chris Grayling and Environment Secretary Andrea Leadsom have all said such a system could help the UK better manage the inflow of people.

But questioned on the policy, Mrs May said: “One of the issues is whether or not points-based systems do work.”

She explained: “A lot of people talk about a points-based system always being the answer in immigration. There is no single silver bullet that is the answer in terms of dealing with immigration.

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“You have to look across the board, you have to look at the whole range of issues. Not just how you bring control through the rules you have for people coming in.

“But also making sure you are rooting out abuse in the system and obviously dealing with people if they are discovered here illegally.”

Mrs May promised last week that curbs on migration would be a key demand of her Brexit negotiations with other EU states.

But asked specifically if EU citizens could be given special treatment as part of some deal to retain access to the single market, the Prime Minister said: “I think what people wanted to see, what came out of the vote, that people wanted to see, was control.

“They wanted to see an ability to be able to control movement of people from the European Union and obviously, that's why I say ‘not free movement as it has been in the past’ and we need to respond to that voice of the British people.”

During the referendum the Leave campaign promised contributions paid to the EU would be diverted to the NHS after Brexit and that VAT on fuel would be scrapped, something they said European regulation prevented.

Vote Leave argued: "It isn't right that unelected bureaucrats in Brussels impose taxes.”

But Mrs May would not commit to working towards either of the pledges when directly challenged, saying only: “I’m going to work for…the best possible deal for the United Kingdom, in terms of the relationship that we have with the EU following is leaving.”

Countries such as Norway gain access to the single market but only by making contributions towards EU funds, something Mr Johnson has signalled he opposes.

Asked if she was prepared to pay any money as part of a deal to secure access to the market, she said: “What we're doing at the moment, is looking at making our preparations before we actually trigger Article 50, going into the formal negotiating process.”

Only last week the Cabinet put on a show of unity at Chequers after reports of a rift between ministers over whether the UK should seek access to the single market.