We study effects of the cross-helicity in the full-sphere large-scale mean-field dynamo models of the $\mathrm{0.3M_{\odot}}$ star rotating with the period of 10 days. In exploring several dynamo scenarios which are stemming from the cross-helicity generation effect, we found that the cross-helicity provide the natural generation mechanisms for the large-scale scale axisymmetric and non-axisymmetric magnetic field. Therefore the rotating stars with convective envelope can produce the large-scale magnetic field generated solely due to the turbulent cross-helicity effect (we call it $\gamma^{2}$-dynamo). Using mean-field models we compare properties of the large-scale magnetic field organization that stem from dynamo mechanisms based on the kinetic (associated with the $\alpha^{2}$ dynamos) and cross-helicity. For the fully convective stars both generation mechanisms can maintain a large-scale dynamos even for the solid body rotation law inside the star. The non-axisymmetric magnetic configurations become preferable when the cross-helicity and the $\alpha$-effect operate independently of each other. This corresponds to situations of the purely $\gamma^{2}$ or $\alpha^{2}$ dynamos. Combination of these scenarios, i.e., the $\gamma^{2}\alpha^{2}$ dynamo can generate preferably axisymmetric, dipole-like magnetic field of strength several kG. Thus we found a new dynamo scenario which is able to generate the axisymmetric magnetic field even in the case of the solid body rotation of the star. We discuss the possible applications of our findings to stellar observations.
