  In this paper, we study the multi-server setting of the \emph{Private
Information Retrieval with Coded Side Information (PIR-CSI)} problem. In this
problem, there are $K$ messages replicated across $N$ servers, and there is a
user who wishes to download one message from the servers without revealing any
information to any server about the identity of the requested message. The user
has a side information which is a linear combination of a subset of $M$
messages in the database. The parameter $M$ is known to all servers in advance,
whereas the indices and the coefficients of the messages in the user's side
information are unknown to any server \emph{a priori}.
  We focus on a class of PIR-CSI schemes, referred to as \emph{server-symmetric
schemes}, in which the queries/answers to/from different servers are symmetric
in structure. We define the \emph{rate} of a PIR-CSI scheme as its minimum
download rate among all problem instances, and define the
\emph{server-symmetric capacity} of the PIR-CSI problem as the supremum of
rates over all server-symmetric PIR-CSI schemes. Our main results are as
follows: (i) when the side information is not a function of the user's
requested message, the capacity is given by
${(1+{1}/{N}+\dots+{1}/{N^{\left\lceil \frac{K}{M+1}\right\rceil -1}})^{-1}}$
for any ${1\leq M\leq K-1}$; and (ii) when the side information is a function
of the user's requested message, the capacity is equal to $1$ for $M=2$ and
$M=K$, and it is equal to ${N}/{(N+1)}$ for any ${3 \leq M \leq K-1}$. The
converse proofs rely on new information-theoretic arguments, and the
achievability schemes are inspired by our recently proposed scheme for
single-server PIR-CSI as well as the Sun-Jafar scheme for multi-server PIR.
