In the first part of this talk, I will speak about shifted quantum affine algebras and their incarnation through the geometry of parabolic Laumon spaces. In the particular case of $\mathfrak{sl}(2)$, we will also see how they naturally give rise to a new family of $3^{n-1}$ modified $q$-Toda systems in type $A_n$.
The second part of the talk is devoted to the definition and key properties of such modified $q$-Toda systems in general, which we define generalizing the original construction of Sevostyanov.
This talk is based on the joint works with M. Finkelberg and R. Gonin.