Chew–Low equations as cremona transformations structure of general intgrals

The Chew–Low equations for the $p$ waves of pion-nucleon scattering with ($3\times3$) crossing symmetry matrix are investigated in the well-known form of a nonlinear system of difference equations. It is shown these equations, interpreted as geometrical transformations, are a special case of Cremona transformations. Using the properties of Cremona transformations, we obtain general functional equations, which depend on three parameters, for algebraic and nonalgebraic invariant curves in the space of solutions of the Chew–Low equations. It is shown that there is only one algebraic invariant curve, a parabola corresponding to the well-known solution. Analysis of the general functional equation for nonalgebraic invariant curves shows that besides this parabola there are three invariant forms which specify implicitly three nonalgebraic curves: a general equation for them is found by fixing the parameters. An important result follows from the transformation properties of these invariant forms with respect to Cremona transformations, namely, the ratio of these forms to appropriate powers is a general integral of the nonlinear system of Chew–Low equations: it is an even antiperiodic function. The structure of a second general integral and the functional equation of which it is a solution are given.