  The three-dimensional characterization of magnetic flux-ropes observed in the
heliosphere has been a challenging task for decades. This is mainly due to the
limitation to infer the 3D global topology and the physical properties from the
1D time series from any spacecraft. To advance our understanding of magnetic
flux-ropes whose configuration departs from the typical stiff geometries, here
we present the analytical solution for a 3D flux-rope model with an arbitrary
cross-section and a toroidal global shape. This constitutes the next level of
complexity following the elliptic-cylindrical (EC) geometry. The mathematical
framework was established by Nieves-Chinchilla et al. (2018) ApJ, with the EC
flux-rope model that describes the magnetic topology with elliptical
cross-section as a first approach to changes in the cross-section. In the
distorted-toroidal flux rope model, the cross-section is described by a general
function. The model is completely described by a non-orthogonal geometry and
the Maxwell equations can be consistently solved to obtain the magnetic field
and relevant physical quantities. As a proof of concept, this model is
generalized in terms of the radial dependence of current density components.
The last part of this paper is dedicated to a specific function,
$F(\varphi)=\delta(1-\lambda\cos\varphi)$, to illustrate possibilities of the
model. This model paves the way to investigate complex distortions of the
magnetic structures in the solar wind. Future investigations will in-depth
explore these distortions by analyzing specific events, the implications in the
physical quantities, such as magnetic fluxes, heliciy or energy, and evaluating
the force balance with the ambient solar wind that allows such distortions.
