  The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable $(C_M,
C_N)$-decomposition of $K_v$ into $\alpha$ $C_M$-factors and $\beta$
$C_N$-factors. We denote a solution to the uniform Hamilton Hamilton-Waterloo
problem by $\hbox{HWP}(v; M, N; \alpha, \beta)$. Our research concentrates on
addressing some of the remaining unresolved cases, which pose a significant
challenge to generalize. We place a particular emphasis on instances where the
$\gcd(M,N)=\{2, 3\}$, with a specific focus on the parameter $M=6$. We
introduce modifications to some known structures, and develop new approaches to
resolving these outstanding challenges in the construction of uniform
$2$-factorizations. This innovative method not only extends the scope of solved
cases, but also contributes to a deeper understanding of the complexity
involved in solving the Hamilton-Waterloo Problem.
