  In this paper we consider the integral orthogonal group with respect to the
quadratic form of signature $(2,3)$ given by $\left(\begin{smallmatrix}
  0 & 1 \\ 1 & 0
  \end{smallmatrix}\right) \perp \left(\begin{smallmatrix}
  0 & 1 \\ 1 & 0
  \end{smallmatrix}\right) \perp (-2N)$ for squarefree $N\in \mathbb{N}$. The
associated Hecke algebra is commutative and the tensor product of its primary
components, which turn out to be polynomial rings over $\mathbb{Z}$ in $2$
algebraically independent elements.
  The integral orthogonal group is isomorphic to the paramodular group of
degree $2$ and level $N$, more precisely to its maximal discrete normal
extension. The results can be reformulated in the paramodular setting by virtue
of an explicit isomorphism. The Hecke algebra of the non-maximal paramodular
group inside $\mathrm{Sp}(2;\mathbb{Q})$ fails to be commutative if $N> 1$.
