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We demonstrate that a spin current flowing through a nanocontact into a uniaxial antiferromagnet with first- and second-order anisotropy can excite a self-localized dynamic magnetic soliton, known as a spin-wave droplet in ferromagnets. The droplet nucleates at a certain threshold current with the frequency of the Néel vector precession laying below the antiferromagnetic resonance. The frequency exhibits nonlinear behavior with the increase in the applied current. At the high value of the applied torque, the soliton mode transforms, and the oscillator emits spin waves propagating in the antiferromagnetic layer. [ABSTRACT FROM AUTHOR]