  In this paper, we prove that any analytic quasi-periodic cocycle close to
constant is the Poincar\'{e} map of an analytic quasi-periodic linear system
close to constant. With this local embedding theorem, we get fruitful new
results. We show that the almost reducibility of an analytic quasi-periodic
linear system is equivalent to the almost reducibility of its corresponding
Poincar\'e cocycle.
  By the local embedding theorem and the equivalence, we transfer the recent
local almost reducibility results of quasi-periodic linear systems \cite{HoY}
to quasi-periodic cocycles, and the global reducibility results of
quasi-periodic cocycles \cite{A,AFK} to quasi-periodic linear systems. Finally,
we give a positive answer to a question of \cite{AFK} and use it to prove
Anderson localization results for long-range quasi-periodic operator with
Liouvillean frequency, which gives a new proof of \cite{AJ05,AJ08,BJ02}. The
method developed in our paper can also be used to prove some nonlinear local
embedding results.
