The Lie point symmetries of the first two equations in the Kadomtsev–Petviashvili (KP) hierarchy, introduced by Jimbo and Miwa, are investigated. The first is the potential KP equation, the second involves four independent variables and is called the Jimbo–Miwa (JM) equation. The joint symmetry algebra for the two equations is shown to have a Kac–Moody–Virasoro structure, whereas the symmetry algebra of the JM equation alone does not. Subgroups of the joint symmetry group are used to perform symmetry reduction and to obtain invariant solutions.