The coupled nonlinear oscillating systems studied do not possess linear normal modes. The nonlinearly related modes of  
m1x¨1+k=1nakx1k+k=1nAk(x1x2)k=0m2x¨2+k=1nbkx2kk=1nAk(x1x2)k=0(1)
 
withk=1,3,5,......
are expressed as superpositions of the linear normal modes of each of the homogeneous systems of the form  
m1x¨1+akx1k+Ak(x1x2)k=0m2x¨2+bkx2kAk(x1x2)k=0(2)
whose force fields are superimposed to yield the force field of equations (1).
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