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Creation of stable intrinsically anisotropic self-bound states with embedded vorticity is a challenging issue. Previously, no such states in Bose-Einstein condensates (BECs) or other physical settings were known. Dipolar BEC suggests a unique possibility to predict stable anisotropic vortex quantum droplets (AVQDs). We demonstrate that they can be created with the vortex' axis oriented perpendicular to the polarization of dipoles. The stability area and characteristics of the AVQDs in the parameter space are revealed by means of analytical and numerical methods. Further, the rotation of the polarizing magnetic field is considered, and the largest angular velocities, up to which spinning AVQDs can follow the rotation in clockwise and anti-clockwise directions, are found. Collisions between moving AVQDs are studied too, demonstrating formation of bound states with a vortex-antivortex-vortex structure. A stability domain for such stationary bound states is identified. A possibility of the creation of AVQDs in a two-component dipolar BEC is briefly considered too.

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To study controlled evolution of nonautonomous matter-wave solitons in spinor Bose-Einstein condensates with spatiotemporal modulation, we focus on a system of three coupled Gross-Pitaevskii (GP) equations with space-time-dependent external potentials and temporally modulated gain/loss distributions. An integrability condition and a nonisospectral Lax pair for the coupled GP equations are obtained. Using it, we derive an infinite set of dynamical invariants, the first two of which are the mass and momentum. The Darboux transform is used to generate one-and two-soliton solutions. Under the action of different external potentials and gain/loss distributions, various solutions for controlled nonautonomous matter-wave solitons of both ferromagnetic and polar types are obtained, such as self-compressed, snake-like and stepwise solitons, and as well as breathers. In particular, the formation of states resembling rogue waves, under the action of a sign-reversible gain-loss distribution, is demonstrated too. Shape-preserving and changing interactions between two nonautonomous matter-wave solitons and bound states of solitons are addressed too. In this context, spin switching arises in the polar-ferromagnetic interaction. Stability of the nonautonomous matter-wave solitons is verified by means of systematic simulations of their perturbed evolution.