Let $P_k$ be the set of solutions of
$$\begin{cases} n + S(n) = m + S(m), \\ kn + S(kn) = km + S(km). \end{cases}$$
We want to map a single solution $(m_0, n_0) \in P_k$, where $10 \nmid m_0, n_0$, to infinite solutions like $(m_1, n_1)$, where $(m_1, n_1) \in P_{k+1}$ or $(m_1, n_1) \in P_{k-1}$ and $10 \nmid m_1, n_1$.
If $(m_0, n_0)$ be a solution in $P_k$, then we claim that $(m_0 + 10^\alpha, n_0 + 10^\alpha)$ is a solution in $P_{k+1}$ for every large $\alpha$. It follows by the same easy calculations and noting that
$$S(n^2 + 2 \times 10^\alpha n + 10^{2\alpha}) = S(n^2) + S(2n) + 1,$$
for all large $\alpha$.
And if $(m_0, n_0)$ is a solution in $P_k$ where $10 \nmid m_0, n_0$, then we replace them by $(10^\alpha - m_0, 10^\alpha - n_0)$ for all large $\alpha$ to reach an infinite number of solutions in $P_{k-1}$. Therefore, it's enough to find a simple solution. For example we have $(9, 12) \in P_0$ and we're done. ■