  Lie algebras of systems of $2 g$ graded heat conduction operators $Q_{2k}$,
where $k = 0,1, \ldots,2 g-1$, determining sigma functions $\sigma(z, \lambda)$
of genus $g = 1,2$, and $3$ hyperelliptic curves are constructed. As a
corollary, it is found that a system of three operators $Q_0, Q_2$ and $Q_4$ is
already sufficient to determine the sigma functions. The operator $Q_0$ is the
Euler operator, and each of the operators $Q_{2k}$, $k>0$, determines a
$g$-dimensional Schr\"odinger equation with quadratic potential in $z$ for a
nonholonomic frame of vector fields in $\mathbb{C}^{2g}$ with coordinates
$\lambda$.
  An analogy of the Cole--Hopf transformation is considered. It associates with
each solution $\varphi(z, \lambda)$ of a linear system of heat equations a
system of nonlinear equations for the vector function $\nabla \ln \varphi(z,
\lambda)$, where $\nabla$ is the gradient of the function in $z$.
  For any solution $\varphi(z, \lambda)$ of the system of heat equations the
graded ring $\mathcal{R}_{\varphi}$ is introduced. It is generated by the
logarithmic derivatives of the function $\varphi(z, \lambda)$ of order of at
least $2$. The Lie algebra of derivations of the ring $\mathcal{R}_{\varphi}$
is presented explicitly. The interrelation of this Lie algebra with the system
of nonlinear equations is shown. In the case when $\varphi(z, \lambda) =
\sigma(z, \lambda)$, this leads to a known result of constructing Lie algebras
of derivations of hyperellitic functions of genus $g = 1,2,3$.
