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

Abstract: 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 Painlev\'e 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 B\"acklund transformations which form the group ${\mathbb Z}^n_2\times{\mathbb Z}^n$.