  This paper studies diagonal implicit symplectic extended
Runge--Kutta--Nystr\"{o}m (ERKN) methods for solving the oscillatory
Hamiltonian system $H(q,p)=\dfrac{1}{2}p^{T}p+\dfrac{1}{2}q^{T}Mq+U(q)$. Based
on symplectic conditions and order conditions, we construct some diagonal
implicit symplectic ERKN methods. The stability of the obtained methods is
discussed. Three numerical experiments are carried out and the numerical
results demonstrate the remarkable numerical behavior of the new diagonal
implicit symplectic methods when applied to the oscillatory Hamiltonian system.
