The aim of this contribution is to refine some of the computations of [6]. The Lindblad equation modelling a two-level dissipative quantum sys- tem is investigated. The control can be interpretated as the action of a laser to rotate a molecule in gas phase, or as the effect of a magnetic field on a spin 1/2 particle. For the energy cost, normal extremals of the maximum principle are solution to a three-dimensional Hamiltonian with parameters. The analy- sis is focussed on an integrable submodel which defines outside singularities a pseudo-Riemannian metric in dimension five. Complete quadratures are given for this subcase by means of Weierstraß elliptic functions. Preliminary com- putations of cut and conjugate loci are also provided for a two-dimensional restriction using [9].