A joint description of the boundary exponent sets of a solution of a linear Pfaff system. I (Q1768980)

We consider completely integrable linear Pfaff system \[ dx^j= \sum A^j_{1k}(t) x^k dt^1+ \sum A^j_{2k}(t) x^k dt^2 \] (\(j,k= 1,\dots, m\); \(i= 1,2\); \(t= (t^1, t^2)\); \(\partial A^j_{1k}/\partial t^2+ \sum A^j_{1l}A^l_{2k}= \partial A^j_{2k}/\partial t^1+ \sum A^j_{2l} A^l_{1k}\)). The upper characteristic vector \(\lambda= \lambda(x)\in\mathbb{R}^2\) of a solution \(x= (x^1,\dots, x^m)\) is defined by \[ (\overline l_x(\lambda))= \limsup{1\over|t|} (\ln|x(t)-(\lambda, t)|= 0,\;\overline l_x(\lambda- \varepsilon e_i)> 0 \] as \(t\to\infty\), where \(\varepsilon> 0\), \(e_i= (2-i-1)\). The upper characteristic exponent \(\overline d=\overline d_x(\lambda)\in \mathbb{R}^2\) is defined by \[ (\overline{\ln}_x(\lambda,\overline d))= \limsup{1\over|\ln t|}(\ln|x(t)|- (\lambda, t)-(\overline d,\ln t))= 0,\;\overline{\ln}_x(\lambda,\overline d-\varepsilon e_i)> 0 \] as \(t\to\infty\), where \(\varepsilon> 0\). The lower characteristic concepts are defined by inferior limits. The authors determine all geometrical properties of these concepts: they construct a smooth Pfaff system if the families of the lower and the upper characteristic exponents are simultaneously prescribed in advance.