New variants of $N=3,4$ superconformal mechanics

We construct superconformal mechanics with $N=3$ and $N=4$ supersymmetries that were inspired by analogies with the supersymmetric Schwarzian mechanics. The Schwarzian, being another system with superconformal symmetry, provides insight into the field content of supersymmetric mechanics, most notably, on the number and properties of the fermionic fields involved. Adding more fermionic fields (four in the $N=3$ case and eight in the $N=4$ case) made it possible to construct systems possessing maximal superconformal symmetries in $N=3$ and $N=4$, namely $OSp(3|2)$ and $D(1,2;\alpha)$. In the case of $N=4$ supersymmetry, we explicitly construct a new variant of $N=4$ superconformal mechanics in which all bosonic subalgebras of $D(1,2;\alpha)$ superalgebra have bosonic realization. In addition, the constructed systems involve $so(3)$ currents whose parametrization is not fixed, which allows to consider different underlying geometries.