Enveloping algebras of restricted Lie superalgebras satisfying non-matrix polynomial identities

Let L be a restricted Lie superalgebra with its enveloping algebra u(L) over a field F of characteristic p>2. A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2\times 2 matrices over F. We characterize L when u(L) satisfies a non-matrix polynomial identity. In particular, we characterize L when u(L) is Lie solvable, Lie nilpotent, or Lie super-nilpotent.