  The Burchnall-Chaundy polynomials $P_n(z)$ are determined by the differential
recurrence relation $$P_{n+1}'(z)P_{n-1}(z)-P_{n+1}(z)P_{n-1}'(z)=P_n(z)^2$$
with $P_{-1}=P_0(z)=1.$ The fact that this recurrence relation has all
solutions polynomial is not obvious and is similar to the integrality of Somos
sequences and the Laurent phenomenon. We discuss this parallel in more detail
and extend it to two difference equations
$$Q_{n+1}(z+1)Q_{n-1}(z)-Q_{n+1}(z)Q_{n-1}(z+1)=Q_n(z)Q_n(z+1)$$ and
$$R_{n+1}(z+1)R_{n-1}(z-1)-R_{n+1}(z-1)R_{n-1}(z+1)=R^2_n(z)$$ related to two
different KdV-type reductions of the Hirota-Miwa and Dodgson octahedral
equations. As a corollary we have a new form of the Burchnall-Chaundy
polynomials in terms of the initial data $P_n(0)$, which is shown to be
Laurent.
