The author considers the existence of a solution for a nonlinear boundary value problem of the form \[ \ddot z_j+ \sum^m_{i=1} b_{ij}(z)\dot z_i\dot z_j= 0,\quad z_j(0)= 0,\quad z_j(1)= 1,\quad j= 1,\dots, m, \] with the additional condition \(0\leq z_j(s)\leq 1\), \(0\leq s\leq 1\), \(j= 1,\dots, m\), where the \(b_{ij}(z)\) are smooth scalar functions, which satisfy the system of differential equations \[ {\partial b_{ij}\over \partial z_k}= b_{ik}\cdot b_{kj},\quad j\neq k,\quad i,j,k= 1,\dots, m. \]