Existence of linear Pfaff systems with lower characteristic set consisting of countably many segments in the space \(\mathbb R^3\) (Q2429683)

The authors consider the Pfaff system \[ \frac{\partial x}{\partial t_i}=A_{i}x,\quad x \in \mathbb R^n,\;t=(t_1,t_2,t_3) \in \mathbb R_+^n,\;i=1,2,3, \tag{1} \] with coefficients matrices \(A_i(t)\) that are bounded and infinitely differentiable in the first octant \( \mathbb R_+^3\) of the space \(\mathbb R^3\) and satisfy there the complete integrable condition \[ \frac{\partial A_i(t)}{\partial t_j}+ A_i(t)A_j(t)= \frac{\partial A_j(t)}{\partial t _i}+ A_j(t)A_i(t),\quad i,j=1,2,3,\;t \in \mathbb R_+^3. \tag{2} \] Then the definition of the lower characteristic set is introduced: Let \(p[x]\) be a lower characteristic vector of a nontrivial solution \(x: \mathbb R_ + ^3 \to \mathbb R^n \backslash \{ 0\}\) or system (1). Thus, \(p[x]\) is determined by the conditions \[ l_x(p[x])\equiv \varliminf_{t \to \infty } \frac{ln\|x(t)\|-(p[x],t)}{\|t\|}=0, \tag{\(3_1\)} \] \[ l_x(p[x]+\varepsilon e_i)< 0,\quad i=1,2,3, \tag{\(3_2\)} \] where the \(e_i\) are the unit coordinate vectors. The union \(P_x=\bigcup p[x]\) of lower characteristic vectors of such a solution is called the lower characteristic set of this solution, and the union \(P(A)=\bigcup P_x\) of the lower characteristic set \(P(A)\) of all nontrivial solutions of the system (1) is called the lower characteristic set of this system. In the present paper the authors prove the existence of a completely integrable linear Pfaff system with the lower characteristic set \(P(A)\) being the union of countable many segments in the space \( \mathbb R_+^3\).