  In a recent paper, Brlek, Jamet and Paquin showed that some extremal infinite
smooth words are also infinite Lyndon words. This result raises a natural
question: are they the only ones? If no, what do the infinite smooth words that
are also Lyndon words look like? In this paper, we give the answer, proving
that the only infinite smooth Lyndon words are $m_{\{a<b\}}$, with $a,b$ even,
$m_{\{1<b\}}$ and $\Delta^{-1}_1(m_{\{1<b\}})$, with $b$ odd, where $m_\A$ is
the minimal infinite smooth word with respect to the lexicographic order over a
numerical alphabet $\A$ and $\Delta$ is the run-length encoding function.
