Finite-dimensional reductions of the discrete Toda chain
  The problem of construction of integrable boundary conditions for the discrete Toda chain is considered. The restricted chains for properly chosen closure conditions are reduced to the well known discrete Painlev\'e equations $dP_{III}$, $dP_{V}$, $dP_{VI}$. Lax representations for these discrete Painlev\'e equations are found. 
T. G. Kazakova