  This paper considers the length of resolution proofs when using
Krishnamurthy's classic symmetry rules. We show that inconsistent linear
equation systems of bounded width over a fixed finite field $\mathbb{F}_p$ with
$p$ a prime have, in their standard encoding as CNFs, polynomial length
resolutions when using the local symmetry rule (SRC-II).
  As a consequence it follows that the multipede instances for the graph
isomorphism problem encoded as CNF formula have polynomial length resolution
proofs. This contrasts exponential lower bounds for
individualization-refinement algorithms on these graphs.
  For the Cai-F\"urer-Immerman graphs, for which Tor\'an showed exponential
lower bounds for resolution proofs (SAT 2013), we also show that already the
global symmetry rule (SRC-I) suffices to allow for polynomial length proofs.
