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In this paper, we seek connections between the Sylvester equation and Kadomtsev–Petviashvili system. By introducing Sylvester equation L M are bold, please chekc if bold neceaasry, if not, please remove all bold of equation − M K = r s T together with an evolution equation set of r and s , master function S (i , j) = s T K j C (I + M C) − 1 L i r is used to construct the Kadomtsev–Petviashvili system, including the Kadomtsev–Petviashvili equation, modified Kadomtsev–Petviashvili equation and Schwarzian Kadomtsev–Petviashvili equation. The matrix M provides τ -function by τ = | I + M C |. With the help of some recurrence relations, the reductions to the Korteweg–de Vries and Boussinesq systems are discussed. [ABSTRACT FROM AUTHOR]