  We study the integrability and equivalence of a generalized Heisenberg
ferromagnet-type equation (GHFE). The different forms of this equation as well
as its reduction are presented. The Lax representation (LR) of the equation is
obtained. We observe that the geometrical and gauge equivalent counterpart of
the GHFE is the modified Camassa-Holm equation (mCHE) with an arbitrary
parameter $\kappa$. Finally, the 1-soliton solution of the GHFE is obtained.
