  It is shown that the relativistic invariance plays a key role in the study of
integrable systems. Using the relativistically invariant sine-Gordon equation,
the Tzitzeica equation, the Toda fields and the second heavenly equation as
dual relations, some continuous and discrete integrable positive hierarchies
such as the potential modified Korteweg-de Vries hierarchy, the potential
Fordy-Gibbons hierarchies, the potential dispersionless
Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL
hierarchy and the second heavenly hierarchies are converted to the integrable
negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy,
the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda
hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the
positive/negative hierarchy dualities are guaranteed by the dualities between
the recursion operators and their inverses. In (2+1)-dimensional cases, the
positive/negative hierarchy dualities are explicitly shown by using the formal
series symmetry approach, the mastersymmetry method and the relativistic
invariance of the duality relations. For the 4-dimensional heavenly system, the
duality problem is studied firstly by formal series symmetry approach. Two
elegant commuting recursion operators of the heavenly equation appear naturally
from the formal series symmetry approach so that the duality problem can also
be studied by means of the recursion operators.
