Symmetries in Connection Preserving Deformations

We wish to show that the root lattice of Bäcklund transformations of the $q$-analogue of the third and fourth Painlevé equations, which is of type $(A_2+ A_1)^{(1)}$, may be expressed as a quotient of the lattice of connection preserving deformations. Furthermore, we will show various directions in the lattice of connection preserving deformations present equivalent evolution equations under suitable transformations. These transformations correspond to the Dynkin diagram automorphisms.

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