The factorization of the flow defined by the Euler-Poisson equations (Q2784963)

The author considers the flow \((p(t),\gamma (t))\) defined by the Euler-Poisson equations NEWLINE\[NEWLINE A\dot p=Ap\times p+\gamma \times r,\quad \dot \gamma =\gamma\times p, NEWLINE\]NEWLINE with \(p=(p_1,p_2,p_3)\in \mathbb{C}^3\), \(\gamma =(\gamma_1,\gamma_2,\gamma_3)\in \mathbb{C}^3\), \(A=\text{diag} (A_1,A_2,A_3)>0\), \(r=(r_1,r_2,r_3)\in \mathbb{R}^3\). The flow has a self-similarity property with respect to the transformation NEWLINE\[NEWLINE (p(t),\gamma (t))\mapsto (\alpha p(\alpha t),\alpha^2\gamma (\alpha t)). NEWLINE\]NEWLINE As a result, the foliation induced by the flow can be mapped onto the foliation of a compact holomorphic manifold. Singular points of the latter foliation are found. Conditions are given for the existence of solutions without singular points.