  This paper is devoted to the one-dimensional heat equation and the non-linear
ordinary differential equations associated to it.
  We consider homogeneous polynomial dynamical systems in the n-dimensional
space, n = 0, 1, 2, .... For any such system our construction matches a
non-linear ordinary differential equation.
  We describe the algorithm that brings the solution of such an equation to a
solution of the heat equation. The classical fundamental solution of the heat
equation corresponds to the case n=0 in terms of our construction. Solutions of
the heat equation defined by the elliptic theta-function lead to the Chazy-3
equation and correspond to the case n=2.
  The group SL(2, C) acts on the space of solutions of the heat equation. We
show this action for each n induces the action of SL(2, C) on the space of
solutions of the corresponding ordinary differential equations. In the case n=2
this leads to the well-known action of this group on the space of solutions of
the Chazy-3 equation. An explicit description of the family of ordinary
differential equations arising in our approach is given.
