Non-autonomous reductions of the KdV equation and multi-component analogs of the Painlevé equations P${}_{34}$ and P${}_3$

We study reductions of the Korteweg–de Vries equation corresponding to stationary equations for symmetries from the noncommutative subalgebra. An equivalent system of $n$ second-order equations is obtained, which reduces to the Painlevé equation P${}_{34}$ for $n=1$. On the singular line $t=0$, a subclass of special solutions is described by a system of $n-1$ second-order equations, equivalent to the P${}_3$ equation for $n=2$. For these systems, we obtain the isomonodromic Lax pairs and Bäcklund transformations which form the group ${\mathbb{Z}}_2^n\times {\mathbb{Z}}^n$.