  In this paper, we develop a new rainbow Hamilton framework, which is of
independent interest, settling the problem proposed by Gupta, Hamann,
M\"{u}yesser, Parczyk, and Sgueglia when $k=3$, and draw the general conclusion
for any $k\geq3$ as follows. A $k$-graph system $\textbf{H}=\{H_i\}_{i\in[n]}$
is a family of not necessarily distinct $k$-graphs on the same $n$-vertex set
$V$, moreover, a $k$-graph $H$ on $V$ is rainbow if $E(H)\subseteq
\bigcup_{i\in[n]}E(H_i)$ and $|E(H)\cap E(H_i)|\leq1$ for $i\in[n]$. We show
that given $\gamma> 0$, sufficiently large $n$ and an $n$-vertex $k$-graph
system $\textbf{H}=\{H_i\}_{i\in[n]}$ , if
$\delta_{k-2}(H_i)\geq(5/9+\gamma)\binom{n}{2}$ for $i\in[n]$ where $k\geq3$,
then there exists a rainbow tight Hamilton cycle. This result implies the
conclusion in a single graph, which was proved by Lang and Sanhueza-Matamala
[$J. Lond. Math. Soc., 2022$], Polcyn, Reiher, R\"{o}dl and Sch\"{u}lke [$J.
Combin. Theory \ Ser. B, 2021$] independently.
