Until recently, the calculation of field-line resonance (FLR) frequencies from magnetic field data, generated by magnetospheric models such as BATSRUS and Tsyganenko (T01), was restricted to orthogonal coordinate systems. With this restriction, only dipolar and axisymmetric configurations are admissible. The matter of addressing more general configurations such as non-axisymmetric stretched and twisted magnetic field topologies requires the use of a non-orthogonal coordinate system. This coordinate system can be constrained by defining the magnetic field as the product of Euler potentials, ${\bf B} = \nabla \alpha_i \times \nabla \alpha _j$, and imposing the condition, $\nabla \cdot {\bf B} = 0$, everywhere. As a consequence, the coordinates, $\alpha_i$, must satisfy the partial differential equation, ${\bf B}\cdot \nabla \alpha_i = 0$. In other words, $\alpha_i$ must be constant along magnetic field lines. Upon solving this differential equation implicitly using known magnetic field intensities, the metric tensor for the resulting basis can be computed. The elements of this tensor can be substituted directly into the eigenvalue problem for general coordinate systems written in covariant notation. The equation for FLR modes has been developed for arbitrary incompressible magnetospheric conditions and has been specialized to the case where spatial variations are constrained along the magnetic field. The result is a fourth-order system of ordinary differential equations, which can be evaluated numerically, provided that the variation of the metric coefficients as a function of the distance along field lines is sufficiently smooth. The eigenvalue problem is solved at several latitudes for a broad range of magnetospheric conditions.