  We investigate the characterization of generators $\mathcal{L}$ of L\'evy
processes satisfying the Liouville theorem: Bounded functions $u$ solving
$\mathcal{L}[u]=0$ are constant. These operators are degenerate elliptic of the
form $\mathcal{L}=\mathcal{L}^{\sigma,b}+\mathcal{L}^\mu$ for some local part
$\mathcal{L}^{\sigma,b}[u]=\text{tr}(\sigma \sigma^{\texttt{T}} D^2u)+b \cdot
Du$ and nonlocal part $$ \mathcal{L}^\mu[u](x)=\int \big(u(x+z)-u(x)-z \cdot
Du(x) \mathbf{1}_{|z| \leq 1}\big) \, \text{d} \mu(z), $$ where $\mu \geq 0$ is
a so-called L\'evy measure possibly unbounded for small $z$. In this paper, we
focus on the pure nonlocal case $\sigma=0$ and $b=0$, where we assume in
addition that $\mu$ is symmetric which corresponds to self-adjoint pure jump
L\'evy operators $\mathcal{L}=\mathcal{L}^\mu$. The case of general L\'evy
operators $\mathcal{L}=\mathcal{L}^{\sigma,b}+\mathcal{L}^\mu$ will be
considered in the forthcoming paper \cite{AlDTEnJa18}. In our setting, we show
that $\mathcal{L}^\mu[u]=0$ if and only if $u$ is periodic wrt the subgroup
generated by the support of $\mu$. Therefore, the Liouville property holds if
and only if this subgroup is dense, and in space dimension $d=1$ there is an
equivalent condition in terms of irrational numbers. In dimension $d \geq 1$,
we have a clearer view of the operators \textit{not} satisfying the Liouville
theorem whose general form is precisely identified. The proofs are based on
arguments of propagation of maximum.
