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We study the compatible left-symmetric conformal algebraic structures on the Lie conformal algebra  W ( a ,  b ), which by definition, is a semidirect sum of the Virasoro Lie conformal algebra and its nontrivial rank one moduleAll compatible left-symmetric conformal algebraic structures on  W ( a ,  b ), under the condition thatis an ideal, are completely determined.

Compatible left-symmetric conformal algebras on W ( a , b )