First integrals of the Maxwell-Bloch system
Consider the three-dimensional polynomial system \[ \frac{dx}{dt}= -ax+y, \quad \frac{dy}{dt}=-by+xz, \quad \frac{dz}{dt}=c(z-\delta_0)y^2-4xy ,\tag{1} \] which is equivalent to the Maxwell-Bloch system in the laser dynamics. The authors prove that system (1) does not possess any rational first integral in the case \(c \neq 0 \) and \( 2\sqrt{(a-b)^2+4\delta_0}/c\) is not an odd number. In the case \(a=b=c=0\), system (1) has two functionally independent polynomial first integrals \[\Phi_1\equiv 2x+z, \quad \Phi_2\equiv 4y+z^2.\] In the case \(a \neq 0, b=c=0\), the unique global analytic first integrals of (1) are of the form \(\Phi(4y^2+z^2)\), where \(\Phi\) is analytic.
scientific article; zbMATH DE number 7226753