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 Schrödinger 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.