Optimal rigid-body angular motions are investigated in the absence of direct control over one of the angular velocity components. A numerical survey of first-order necessary conditions for optimally reveals that, over a range of boundary conditions, there are, in general, several distinct extremal solutions. A classification in terms of subfamilies of extremal solutions is presented. Domains of existence of the extremal subfamilies are established. Locus of Darboux points are obtained, and global optimality of extremal solutions is observed in relation to Darboux points. Local optimality for the candidate minimizers is verified by investigating the second-order necessary conditions. TUDIES of optimal flight generally employ dynamic models based on point-mass equations of motion. Indeed, for many purposes these models must be further simplified, leading to energy models and the like. Recently there has been considerable interest in highly agile or supermaneuver able aircraft. Maneuvers of interest in these applications include rapid fuselage pointing and suggest the need to include rigidbody rotational dynamics in optimal flight studies. The usual flight mechanics wisdom is that the rigid-body rotations are fast compared with translational motions (short-period visa-vis phugoid frequencies). This suggests a singular perturbation scheme to divide and conquer these problems. However, some of the maneuvers discovered in simulator studies are delicately choreographed so that considerable care is needed in choice of coordinates and in time-scale selection. The key point is that careful analysis is needed to sort out various effects and to gain insight into these supermaneuvers. The present study is motivated by the problem of angular velocity control for a supermaneuvera ble aircraft. In one scenario of interest, primarily the low-dynamic-pressure regime at high angles of attack, the usual aerodynamic control surfaces are ineffective for generating control moments. In such flight regimes, it is proposed that thrust-vectored propulsive controls be uspd as the primary source of control moments, which may be generated by deflecting vanes in the exhaust flowfield of a jet-powered aircraft. Owing to the location of engines, we anticipate that the rolling moment (in the body-fixed axes) that can be generated is going to be small in comparison to the pitching and the yawing moments. As an extreme case, we consider the case when the rolling moment is identically zero. Hence any rolling motion is controlled indirectly through the pitch and yaw motions. Thus we are motivated to study optimal rigid-body rotational maneuvers in the absence of rolling moments.