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In this paper, under some natural condition, a complete classification ofcompatible left-symmetric conformal algebraic structures on the Lie conformalalgebra $\mathcal{W}(a,b)$ is presented. Moreover, applying this result, weobtain a class of compatible left-symmetric algebraic structures on thecoefficient algebra of $\mathcal{W}(a,b)$.

Classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra $\mathcal{W}(a,b)$