Bäcklund transformations for a \(\varvec{(3+1)}\) -dimensional \(\varvec{P-}\) type evolution equation are constructed using the truncated Painlevé expansion and associated nonlocal symmetries. This obtained nonlocal symmetry is localized into the classical Lie symmetry by first extending the considered equation to an enlarged form and then introducing appropriate potential variables. On the basis of the consistent tanh expansion method (CTE), this equation is shown to be CTE-solvable and some classes of interaction solutions are derived. In addition, exact solitary waves are derived using a symbolic computation approach.
