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* Portuguese nets twice in 2-1 success
* Real climb above Atletico into second
* Atletico beaten 1-0 at home to Sociedad (Adds Atletico result, quotes)
By Iain Rogers
MADRID, March 10 (Reuters) - Real Madrid followed up Tuesday's Champions League success at Manchester United by climbing above beaten city rivals Atletico into second in La Liga when Cristiano Ronaldo struck twice in a 2-1 win at second-bottom Celta Vigo on Sunday.
Ronaldo netted the goal that sent Real through to the last eight of Europe's elite club competition at the expense of his former club and his latest double took his league tally for the season to 26, 14 behind top scorer Lionel Messi of Barcelona.
The victory was far from convincing but put champions Real a point clear of Atletico on 58 points with 11 games left, 13 behind leaders Barca who won 2-0 at basement side Deportivo La Coruna on Saturday.
Atletico had won all 13 home matches this season but went down 1-0 to Real Sociedad in the late kickoff.
Real have little chance of denying Barca a fourth title in five seasons but centre back Pepe said they would keep fighting.
"It is tough to challenge for La Liga because they are a lot of points in front of us but we have to maintain our motivation and try to honour the shirt," the Portuguese told reporters.
Celta are fighting to avoid an immediate return to the second division after winning promotion last term and started brightly at their Balaidos stadium.
Ronaldo nodded against the crossbar in a goalless first half before the Portugal forward pounced on a loose ball and fired in from close range in the 61st minute.
Celta levelled two minutes later when an Iago Aspas strike deflected off Pepe and bounced past Diego Lopez into the net before Kaka won a 71st-minute penalty when he was felled by Lopez's opposite number Javi Varas.
Ronaldo dispatched the spot-kick for his 19th goal in all competitions since the turn of the year, three more than Messi.
Real rode their luck as the rain began to lash down in the closing stages and Celta substitute Park Chu-young nodded against the bar a minute from time.
Atletico coach Diego Simeone has transformed the club where he had two stints as a player since taking over midway through last season and agreed a new contract this week until the end of the 2016-17 campaign.
He had to watch from the technical area as his side's winning home win was ended by a well-organised Sociedad side who scored the only goal of the game eight minutes after halftime.
Xabi Prieto raced clear on the right and his powerful low shot flew into the net between the legs of onrushing Atletico goalkeeper Thibaut Courtois.
The victory lifted San Sebastian-based Sociedad to fifth, level on 44 points with fourth-placed Malaga and looking good for a place in Europe next season.
Valencia missed a chance to claim fourth spot when they were beaten 1-0 at Athletic Bilbao earlier on Sunday.
Eliminated from the Champions League at Paris St Germain on Wednesday, the defeat dented the cash-strapped club's chances of securing a lucrative place in the competition for next season.
Bilbao forward Iker Muniain clipped home a loose ball in the 80th minute to put the home side ahead at a vibrant San Mames stadium. Valencia substitute Roberto Soldado had a chance to level minutes later but somehow scooped the ball over the bar with the goal at his mercy.
The defeat left Ernesto Valverde's side in seventh on 42 points, one behind Betis and two adrift of Malaga and Sociedad.
Bilbao, who have had a miserable season after reaching the finals of the Europa League and the King's Cup last term, are 14th on 32 points.
Levante, through to the last 16 of the Europa League, are also bidding for a return to continental competition next season but were held to a 0-0 draw at home to Getafe.
Valencia-based Levante are 10th on 36 points, three behind Madrid side Getafe. (Editing by Ed Osmond)