Non-trivial stably free modules over crossed products

We consider the class of crossed products of Noetherian domains with universal enveloping algebras of Lie algebras. For algebras from this class we give a sufficient condition for the existence of projective non-free modules. This class includes Weyl algebras and universal envelopings of Lie algebras, for which this question, known as the non-commutative Serre's problem, has been extensively studied previously. It turns out that the method of lifting of non-trivial stably free modules from simple Ore extensions can be applied to crossed products after an appropriate choice of filtration. The motivating examples of crossed products are provided by the class of relativistic internal time algebras, originating in non-equilibrium physics.