  The Ward equation, also called the modified 2+1 chiral model, is obtained by
a dimension reduction and a gauge fixing from the self-dual Yang-Mills field
equation on $R^{2,2}$. It has a Lax pair and is an integrable system. Ward
constructed solitons whose extended solutions have distinct simple poles. He
also used a limiting method to construct 2-solitons whose extended solutions
have a double pole. Ioannidou and Zakrzewski, and Anand constructed more
soliton solutions whose extended solutions have a double or triple pole. Some
of the main results of this paper are: (i) We construct algebraic B\"acklund
transformations (BTs) that generate new solutions of the Ward equation from a
given one by an algebraic method. (ii) We use an order $k$ limiting method and
algebraic BTs to construct explicit Ward solitons, whose extended solutions
have arbitrary poles and multiplicities. (iii) We prove that our construction
gives all solitons of the Ward equation explicitly and the entries of Ward
solitons must be rational functions in $x, y$ and $t$. (iv) Since stationary
Ward solitons are unitons, our method also gives an explicit construction of
all $k$-unitons from finitely many rational maps from $C$ to $C^n$.
