  Eigenfunctions of integrable planar billiards are studied - in particular,
the number of nodal domains, $\nu$, of the eigenfunctions are considered. The
billiards for which the time-independent Schr\"odinger equation (Helmholtz
equation) is separable admit trivial expressions for the number of domains.
Here, we discover that for all separable and non-separable integrable
billiards, $\nu $ satisfies certain difference equations. This has been
possible because the eigenfunctions can be classified in families labelled by
the same value of $m\mod kn$, given a particular $k$, for a set of quantum
numbers, $m, n$. Further, we observe that the patterns in a family are similar
and the algebraic representation of the geometrical nodal patterns is found.
Instances of this representation are explained in detail to understand the
beauty of the patterns. This paper therefore presents a mathematical connection
between integrable systems and difference equations.
