Amplitude analyses of the decays $\chi_c1 -> \eta \pi^+ \pi^-$ and $\chi_c1 -> \eta' \pi^+ \pi^-$

Using a data sample of 2: 59 x 10(7) psi(2S) decays obtained with the CLEO-c detector, we perform amplitude analyses of the complementary decay chains psi(2S) -> gamma chi(c1); chi(c1) -> eta pi(+)pi(-) and psi(2S) -> gamma chi(c1); chi(c1) -> eta 'pi(+)pi(-). We find evidence for an exotic P-wave eta 'pi amplitude, which, if interpreted as a resonance, would have parameters consistent with the pi(1)(1600) state reported in other production mechanisms. We also make the first observation of the decay a(0)(980) -> eta 'pi and measure the ratio of branching fractions B(a(0)(980) -> eta 'pi/B(a(0)(980) -> eta pi = 0.064 +/- 0.014 +/- 0.014. The pi pi spectrum produced with a recoiling eta is compared to that with eta ' recoil.