Ordinary differential equations associated with the heat equation
  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. 
Victor M. Buchstaber, Elena Yu. Bunkova
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