  In a 2004 paper by V. M. Buchstaber and D. V. Leykin, published in
"Functional Analysis and Its Applications," for each $g > 0$, a system of $2g$
multidimensional heat equations in a nonholonomic frame was constructed. The
sigma function of the universal hyperelliptic curve of genus $g$ is a solution
of this system. In the work arXiv:2007.08966 explicit expressions for the
Schr\"odinger operators that define the equations of the system considered were
obtained in the hyperelliptic case.
  In this work we use these results to show that if the initial condition of
the system considered is polynomial, then the solution of the system is
uniquely determined up to a constant factor. This has important applications in
the well-known problem of series expansion for the hyperelliptic sigma
function. We give an explicit description of the connection of such solutions
to well-known Burchnall-Chaundy polynomials and Adler-Moser polynomials. We
find a system of linear second-order differential equations that determines the
corresponding Adler-Moser polynomial.
