The parabolic algebra A_p is the weakly closed operator algebra on L^2(R) generated by the unitary semigroup of right translations and the unitary semigroup of multiplication by the analytic exponential functions e^{iλx}, λ ≥ 0. It is reflexive, with an invariant subspace lattice Lat A_p which is naturally homeomorphic to the unit disc (Katavolos and Power, 1997). The structure of Lat A_p is used to classify strongly irreducible isometric representations of the partial Weyl commutation relations. A formal generalisation of Arveson's notion of a synthetic commutative subspace lattice is given for general subspace lattices, and it is shown that Lat A_p is not synthetic relative to the H∞(R) subalgebra of A_p. Also, various new operator algebras, derived from isometric representations and from compact perturbations of A_p, are defined and identified.