On Hamiltonian Formalism for Dressing Chain Equations of Even Periodicity

We propose a Hamiltonian formalism for $N$ periodic dressing chain with the even number $N$. The formalism is based on Dirac reduction applied to the $N+1$ periodic dressing chain with the odd number $N+1$ for which the Hamiltonian formalism is well known. The Hamilton dressing chain equations in the $N$ even case depend explicitly on a pair of conjugated Dirac constraints and are equivalent to $A^{(1)}_{N-1}$ invariant symmetric Painlev\'e equations.