  In this paper, we mainly predict the rates of $M$1 radiative and spin-nonflip
$\pi\pi$ transitions of the $B_{c}$-meson under the nonrelativistic Cornell
potential model with a screening potential effect. We employ the numerical wave
function to determine the $M$1 radiative transition widths of $B_c$ excited
states and utilize the Kuang-Yan proposed method for the spin-nonflip $\pi\pi$
transitions among $B_c$ states. Our theoretical results are valuable for
studying the $M$1 radiative and spin-nonflip $\pi\pi$ transition processes of
$B_c$ states in experiments.
