The effect of molecular reorientation and energy migration on the steady-state and time-resolved fluorescence anisotropies is investigated. The energy migration takes place between isotropically oriented donors in a two-dimensional disordered system. The donor molecules are 2,5,8,11-tetra-tert-butylperylene (TBPe) which are solubilized in the hydrophobic part of unilamellar vesicles. The vesicles are prepared with 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) in a glycerol-water solution. At low number density of TBPe the fluorescence anisotropies are solely due to the reorientation of TBPe. The rotational motion of TBPe can be described as that of an oblate ellipsoid in a solvent. At high number densities of TBPe an additional depolarization, due to electronic energy migration, is observed. The fluorescence relaxation is single exponential at all temperatures (240 and 320 K) and concentrations studied. The order parameter of TBPe in the DOPC could be determined at temperatures above 260 K and was found to be small and constant. The time-resolved fluorescence anisotropies measured, at high and low number densities of TBPe, were analyzed globally. The analytical model used is based on the two-particle (tp) approximation of energy migration in a two-dimensional system. It was shown that the effect of vesicle curvature is negligible. The model contains only one unknown parameter, namely, a factor K which is modulated by molecular reorientations. In the so-called "dynamic" and "static" limits, i.e., for very fast or slow rotational rates, the tp model predicts the value of K(dyn) = 0.874 and K(stat) = 0.740, respectively. From experiments we find that the values of K agree reasonably with K(stat) at temperatures between 260 and 320 K, while it surprisingly approaches the value of K(dyn) at lower temperatures. The steady-state anisotropy was calculated by using the tp model. A good agreement with the experimental anisotropy was obtained for K = K(stat) at temperatures above 260 K. The increase of K observed at lower temperatures coincides with the phase transition of DOPC in the range 240-260 K. Thereby, TBPe molecules probably reorient and/or become spatially or orientationally correlated.