  A two-component-two-dimensional coupled with one-component-three-dimensional
(2C2Dcw1C3D) flow may also be called a real Schur flow (RSF), as its velocity
gradient is uniformly of real Schur form, the latter being the intrinsic local
property of any general flows. The thermodynamic and `vortic' fine structures
of RSF are exposed and, in particular, the complete set of equations governing
a (viscous and/or driven) 2C2Dcw1C3D flow are derived. The Lie invariances of
the decomposed vorticity 2-forms of RSFs in $d$-dimensional Euclidean space
$\mathbb{E}^d$ for any interger $d\ge 3$ are also proven, and many
Lie-invariant fine results, such as those of the combinations of the entropic
and vortic quantities, including the invariances of the decomposed Ertel
potential vorticity (and their multiplications by any interger powers of
entropy) 3-forms, then follow.
