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We consider the class of polynomial differential systems of the form 
$\dot{x} =\,\lambda x\,-\,y\,+\,{{P}_{n}}\left( x,\,y \right)$
, 
$\dot{y} =\,x\,+\,\lambda y\,+\,{{Q}_{n}}\left( x,\,y \right)$
where 
${{P}_{n}}$
and 
${{Q}_{n}}$
are homogeneous polynomials of degree 
$n$
. For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence.
