  For each $\lambda \in \mathbb N^*$, we consider the integral equation: \[
\int_{\lambda y} ^{\lambda x} f(t)\, d t = f(x) - f(y) \mbox{ for every
$(x,y)\in {\mathbb R}_+^2$,} \] where $f$ is the concatenation of two
continuous functions $f_a,f_b:[0,\lambda] \rightarrow {\mathbb R}$ along a word
$u= u_0u_1\cdots\in\{a,b\}^{\mathbb N}$ such that $u=\sigma(u)$, where $\sigma$
is a $\lambda$-uniform substitution satisfying some combinatorial conditions.
  There exists some non-trivial solutions. We show in this work that the
dimension of the set of solutions is at most two.
