Invariant algebraic surfaces and algebraic first integrals of the Maxwell-Bloch system (Q2331534)

The Maxwell-Bloch system \[ \dot x=-ax+y,\ \dot y=-cy+xz,\ \dot z=-b(z-d)-4xy\tag{\(*\)} \] modeling laser systems is studied in case \(a=0\). In that case, system (\(*\)) can be transformed into the form \[ x'=y,\ y'=cy-kz+dx,\ z'=xy+bz,\tag{\(**\)} \] where \(b,c,d\) are real parameters. The author proves sufficient and necessary conditions by means of the parameters \(b,c,d\) for system (\(**\)) \begin{itemize} \item[(i)] to be algebraically integrable \((b=c=0)\). \item[(ii)] to have a polynomial first integral \((b=0\) or \(b=-2c\ne 0\), \(d=0)\). \item[(iii)] to have an invariant algebraic surface \((b=c\ne 0,d=0)\). \end{itemize}