Meromorphy of solutions for a system of $N$ equations of Painlevé 34 type related to negative symmetries of the Korteweg–de Vries equation

We prove the property of meromorphic extendability for every local holomorphic solution of a system of nonlinear nonautonomous ordinary differential equations. This system is a vector generalization of Painlevé's 34 equation (which is in its turn equivalent to the second Painlevé equation) and coincides with the stationary part of a symmetry of the Korteweg–de Vries equation obtained as the sum of the stationary parts of the classical Galilean symmetry and $N$ negative symmetries of this integrable evolutionary equation.