We use the recently established rationality of correlation functions in a globally conformally invariant quantum field theory satisfying Wightman axioms to construct a family of solvable models in the four-dimensional Minkowski space–time. We consider the model of a neutral two-dimension scalar field φ in detail. It depends on a positive real parameter c, an analogue of the Virasoro central charge; for all (finite) c, it admits an infinite number of conserved symmetric tensor currents. The operator product algebra of φ coincides with a simpler one generated by a bilocal scalar field V(x 1,x 2) of dimension 1+1. The modes of V together with the unit operator span an infinite-dimensional Lie algebra ℒ V , whose vacuum (i.e., zero-energy lowest-weight) representations depend only on the central charge c. The Wightman positivity (i.e., unitarity of the representations of ℒ V ) is proved equivalent to c∈ℕ.