  Vector modes represent the most general state of light in which, the spatial
and polarisation degrees of freedom are coupled in a non-separable way.
Crucially, while polarisation is limited to a bi-dimensional space, the spatial
degree of freedom can take any spatial profile. However, most generation and
application techniques are mainly limited to spatial modes with polar
cylindrical symmetry, such as Laguerre- and Bessel-Gauss modes. In this
manuscript we put forward a novel class of vector modes with its spatial degree
of freedom encoded in the set of helical Mathieu-Gauss beams of the elliptical
cylindrical coordinates. We first introduce these modes theoretically and
outline their geometric representation on the higher-order Poincar\'e sphere.
Later on, we demonstrate their experimental generation using a
polarisation-insensitive technique comprising the use of a digital micromirror
device. Finally, we provide with a qualitative and a quantitative
characterisation of the same using modern approaches based on quantum mechanics
tools. It is worth mentioning that non-polar vector beams are highly desired in
various applications, such as optical trapping and optical communications.
