Method of local construction of invariant subspaces in the solution space of Chew–Low equations

A nonlinear system offunctlonal equations for the matrix elements of the $S$-matrix is formulated on the basis of Chew–Low equations. A transition is made to projective coordinates in the spade of the matrix elements of the $S$-matrix and the unitarity condiflons are linearlzed. On the basis of a geometrical interpretation of the system of nonlinear functional equations as a transformation in an $(n-1)$-dimensional real space, it is shown that some of the solutions of the original system of equations lie on invariant hypersurfaecs of this space. A method is proposed for the local construction of the invariant hypersurfaces in the neighborhood of the fixed points of the transformation. This method is applied to the Chew–Low equations with $3\times 3$ and $4\times 4$ crossing matrices. It is shown that, if the Chew–Low equations have a selution, the arbitrariness, which is a generalization of the well-known $\beta$-arbitrariness, in the solutions of the class considered is not exhaustive.