New integrable problems in a rigid body dynamics with cubic integral in velocities

• A new multi-parameters integrable system with a cubic integral is introduced. • We present new integrable problems of a particle in the plane. • We introduce new integrable problems of a particle on Pseudo sphere. • A rare integrable problem in a rigid body-gyrostat is presented. We introduce a new family of the 2D integrable mechanical system possessing an additional integral of the third degree in velocities. This system contains 20 arbitrary parameters. We also clarify that the majority of the previous systems with a cubic integral can be reconstructed from it as a special version for certain values of those parameters. The applications of this system are extended to include the problem of motion of a particle and rigid body about its fixed point. We announce new integrable problems describing the motion of a particle in the plane, pseudosphere, and surfaces of variable curvature. We also present a new integrable problem in a rigid body dynamics and this problem generalizes some of the previous results for Sokolov-Tsiganov, Yehia, Stretensky, and Goriachev.