Bosonic symmetries of the extended fermionic $(2N,2M)$-Toda hierarchy

We construct the additional symmetries of the fermionic $(2N,2M)$-Toda hierarchy based on a generalization of the $N=(1|1)$ supersymmetric two-dimensional Toda lattice hierarchy. These additional flows constitute a $w_\infty\times w_\infty$ Lie algebra. As a bosonic reduction of the $N=(1|1)$ supersymmetric two-dimensional Toda lattice hierarchy and the fermionic $(2N,2M)$-Toda hierarchy, we define a new extended fermionic $(2N,2M)$-Toda hierarchy that admits a bosonic Block-type superconformal structure.