We study a fluid of nematogenic molecules with centres of mass constrained to lie in a plane but with axes free to rotate in any direction. An external disorienting field perpendicular to the plane along with a second orienting field in the plane induce an in-plane order–disorder transition. We analyse the behaviour of this simple biaxial model using a well-established generalization of molecular integral equation methods built upon specially tailored basis functions that maintain orthogonality in the presence of anisotropy. Computer simulation and integral equation calculations predict an isotropic–nematic transition at low temperatures in zero field and an in-plane transition at somewhat higher temperatures in the presence of the disorienting field. The oriented states obtained in the presence of both fields can subsequently be used as input to uncover in detail first the transition in the absence of the in-plane orienting field and finally the spontaneous transition in the absence of any field. According to the simulation, the transition apparently belongs to the Berezinskii–Kosterlitz–Thouless defect-mediated type, whereas the theory reproduces a weak first-order transition.