In this paper, we discuss the existence and regularity of solutions of the following strongly indefinite systems involving fractional elliptic operators on a smooth bounded domain W in

where  refer to any of the two types of operators  or    and m are fixed real numbers and

·  for the spectral fractional Laplace operator

·  for the restricted fractional Laplace operator

fractional Laplace operator, fractional elliptic systems, critical hyperbole.