Liftable vector fields over corank one multigerms

In this paper, a systematic method is given to construct all liftable vector fields over an analytic multigerm $f: (\mathbb{K}^n, S)\to (\mathbb{K}^p,0)$ of corank at most one admitting a one-parameter stable unfolding.
Comment: 34 pages. In ver. 2, several careless mistakes for calculations in Section 6 were corrected