For each $λ\in \mathbb N^*$, we consider the integral equation: \[ \int_{λy} ^{λx} f(t)\, d t = f(x) - f(y) \mbox{ for every $(x,y)\in {\mathbb R}_+^2$,} \] where $f$ is the concatenation of two continuous functions $f_a,f_b:[0,λ] \rightarrow {\mathbb R}$ along a word $u= u_0u_1\cdots\in\{a,b\}^{\mathbb N}$ such that $u=σ(u)$, where $σ$ is a $λ$-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.