| Abstract: |
| We consider existence and nonexistence of first integrals near integral curves with finite time for autonomous dynamical systems. We characterize how many first integrals can exist near periodic orbits. This characterization is an improvement of Poincar\`{e}`s classical criterion using the variational equation near periodic orbits. Moreover, we show that there is a flow-box coordinate for non-periodic integral curves with finite time. |
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