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We present a new ray bending approach, referred to as the Eigenray method, for solving two-point boundary-value kinematic and dynamic ray tracing problems in 3D smooth heterogeneous general anisotropic elastic media. The proposed Eigenray method is aimed to provide reliable stationary ray path solutions and their dynamic characteristics, in cases where convergence to the stationary paths, based on conventional initial-value ray shooting methods, becomes challenging. The kinematic ray tracing solution corresponds to the vanishing first traveltime variation, leading to a stationary path, and is governed by the nonlinear second-order Euler-Lagrange equation (Part I). In Part II we elaborate on theoretical aspects of the proposed method and validate its correctness for general anisotropy. In Part III we use a finite-element approach, applying the weak formulation. In Part IV we propose an efficient method to compute the geometric spreading of the entire stationary ray path using the global traveltime Hessian. In Part V we formulate the dynamic ray tracing, considering the second traveltime variation, which leads to the linear second-order Jacobi equation, and in Part VI we relate the proposed Lagrangian approach to the commonly used Hamiltonian approach. The solution is provided in Part VII, where we implement a similar finite element approach applied for the kinematic problem. In both kinematic and dynamic problems, in between the nodes, the values of the ray characteristics are computed with the Hermite interpolation, which we find most natural for anisotropic media. We distinguish two types of stationary rays, delivering either a minimum or a saddle-point traveltime (due to caustics).