Integrability of the constrained rigid body

The integrability theory for the differential equations, which describe the motion of an unconstrained rigid body around a fixed point is well known. When there are constraints the theory of integrability is incomplete. The main objective of this paper is to analyze the integrability of the equations of motion of a constrained rigid body around a fixed point in a force field with potential U(gamma)=U(gamma(1),gamma(2),gamma(3)). This motion subject to the constraint <nu,omega >=0 with nu is a constant vector is known as the Suslov problem, and when nu=gamma is the known Veselova problem, here omega=(omega(1),omega(2),omega(3)) is the angular velocity and <,> is the inner product of R-3. We provide the following new integrable cases. (i) The Suslov's problem is integrable under the assumption that nu is an eigenvector of the inertial tensor I and the potential is such that U=-1/2I1I2 (I-1 mu(2)(1)+I-2 mu(2)(2)), where I-1,I-2, and I-3 are the principal moments of inertia of the body, mu(1) and mu(2) are solutions of the first-order partial differential equation gamma(3)(partial derivative mu(1)/partial derivative gamma(2)-partial derivative mu(2)/partial derivative gamma(1))-gamma(2) partial derivative mu(1)/partial derivative gamma(3)+gamma(1) partial derivative mu(2)/partial derivative gamma(3)=0. (ii) The Veselova problem is integrable for the potential U=-Psi(2)(1)+Psi(2)(2)/2(I-1 gamma(2)(2)+I-2 gamma(2)(1)), where Psi(1) and Psi(2) are the solutions of the first-order partial differential equation (I-2-I-1)gamma(1 gamma 2) <gamma,partial derivative Psi(2)/partial derivative gamma >+I-1 gamma(2) partial derivative Psi(2)/partial derivative gamma(1)-I-2 gamma(1) partial derivative Psi(2)/partial derivative gamma(2) - p(gamma 3 <gamma,partial derivative Psi(1)/partial derivative gamma >-partial derivative Psi(1)/partial derivative gamma(3)=0, where p root I1I2I3(gamma(2)(1)/I-1+gamma(2)(2)/I-2+gamma(2)(3)/I-3). Also it is integrable when the potential U is a solution of the second-order partial differential equation 2 partial derivative U/partial derivative tau(3)+I1I2I3 partial derivative U-2/partial derivative tau(2)(2) + (tau(2)-I-1-I-2-I-3) partial derivative U-2/partial derivative tau(3)partial derivative tau(2)+tau(3) partial derivative U-2/partial derivative tau(2)(3)=0, where tau(2) = I-1 gamma(2)(1)+I-2 gamma(2)(2)+I-3 gamma(2)(3) and tau(3) = gamma(2)(1)/I-1+gamma(2)(2)/I-2+gamma(2)(3)/I-3. Moreover, we show that these integrable cases contain as a particular case the previous known results.