The second-order ordinary differential equation (ODE) often emerges from applied science, such as orbital mechanics, quantum mechanics, physical chemistry, electronics. As we all know, Runge-Kutta-Nyström (RKN) method is indispensable when solving the second-order ODE. In addition, there are also some intrinsic properties in these fields. How to preserve these properties must be considered when seeking the numerical solutions. Thus, in this paper, we focus on the construction of the implicit RKN method. Combining the symmetry conditions and symplecticness conditions, sixth-order implicit exponentially fitted/trigonometrically fitted RKN integrators are obtained. The designed methods have the power of solving Hamiltonian system. And we make some numerical experiments to show the efficiency and competence of the new methods compared with some highly efficient implicit codes in the literature.

二阶常微分方程 (ODE) 经常出现在应用科学中，如轨道力学、量子力学、物理化学、电子学。众所周知，Runge-Kutta-Nyström (RKN) 方法在求解二阶 ODE 时必不可少。此外，这些领域也有一些固有的特性。在寻求数值解时必须考虑如何保持这些性质。因此，在本文中，我们专注于隐式 RKN 方法的构建。结合对称性条件和辛性条件，得到六阶隐式指数拟合/三角拟合RKN积分器。所设计的方法具有求解哈密顿系统的能力。