  We formulate a comprehensive hydrodynamic theory of two-dimensional liquid
crystals with generic $p-$fold rotational symmetry, also known as $p-$atics, of
which mematics $(p=2)$ and hexatics $(p=6)$ are the two best known examples.
Previous hydrodynamic theories of $p-$atics are characrerized by continuous
${\rm O}(2)$ rotational symmetry, which is higher than the discrete rotational
symmetry of $p-$atic phases. By contrast, here we demonstrate that the discrete
rotational symmetry allows the inclusion of additional terms in the
hydrodynamic equations, which, in turn, lead to novel phenomena, such as the
possibility of flow alignment at high shear rates, even for $p>2$. Furthermore,
we show that any finite imposed shear will induce long-ranged orientational
order in any $p-$atic liquid crystal, in contrast to the quasi-long-ranged
order that occurs in the absence of shear. The induced order parameter scales
like a non-universal power of the applied shear rate at small shear rates.
