Dynamics of $CH_2CI$ group in ${alpha},2,6$-trichlorotoluene dissolved in $CDCl_3$ was studied by observing various relaxation modes for $^{13}C$ under proton undecoupled condition. Partially relaxed $^{13}C$ spectra were obtained at $34^{circ}C$ as a function of evolution time after applying various designed pulse sequences to this $AX_2$ spin system. It was found that nonlinear regression analysis of the relaxation data for these magnetization modes could provide the information about dipolar and spin-rotational auto-correlation and cross-correlation spectral densities for fluctuation of the $^{13}C-^1H$ internuclear vector in $CH_2Cl$ group. The results show that the effect of cross-correlation is comparable in magnitude to that of auto-correlation and the relaxation in this spin system is dominated by dipolar mechanism rather than spin-rotational one. From the resulting spectral density data we could calculate the bond angle ${angle}HCH;(105.1$^{circ}$) and elements of the rotational diffusion tensor for $CH_2Cl$ group.