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We present an approximate nonlinear theory of microwave generation by spin-polarized direct current in a magnetic nanocontact magnetized in an arbitrary direction. We argue that, when the spin-transfer torque caused by spin-polarized current compensates the natural magnetic dissipation in a "free" layer of the nanocontact, a nonlinear quasi-uniform precession of magnetization about the direction of the internal bias magnetic field is excited. With the increase of the current magnitude the angle of precession increases, making precession strongly nonlinear and reducing the projection Mz of the precessing magnetization vector on the precession axis (z axis). This reduction of Mz is responsible for the nonlinear limitation of the precession amplitude and for the nonlinear frequency shifts of the generated microwave oscillations. Because of the influence of demagnetizing fields in the "free" layer, the nonlinear frequency shifts have different magnitudes and signs for different orientations of the external bias field He. The theory gives a good qualitative, and even partly quantitative, explanation of the main part of microwave magnetization dynamics experimentally observed in magnetic nanocontacts.