  We show that there is non-uniqueness for the Calder{\'o}n problem with
partial data for Riemannian metrics with H{\"o}lder continuous coefficients in
dimension greater or equal than three. We provide simple counterexamples in the
case of cylindrical Riemannian manifolds with boundary having two ends. The
coefficients of these metrics are smooth in the interior of the manifold and
are only H{\"o}lder continuous of order $$\rho$\<1$ at the end where the
measurements are made. More precisely, we construct a toroidal ring $(M, g)$
which is not a warped product manifold, and we show that there exist in the
conformal class of g an infinite number of Riemannian $\tilde{g} = c^4 g such
that their corresponding partial Dirichlet-to-Neumann maps at one end coincide.
The corresponding smooth conformal factors are harmonic with respect to the
metric g and do not satisfy the unique continuation principle.
