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The "\tau\to 3\pi\eta\nu" decay with the "\eta \to \gamma\gamma" mode isstudied using 384 fb^-1 of data collected by the BaBar detector. The branchingfraction is measured to be (1.60+/-0.05+/-0.11)x10^-4. It is found that "\tau\to f1 \pi \nu to 3\pi \eta \nu" is the dominant decay mode with a branchingfraction of (1.11+/-0.06+/-0.05)x10^-4. The first error on the branchingfractions is statistical and the second systematic. In addition, a 90%confidence level upper limit on the branching fraction of the "\tau \to\eta'(958) \pi \nu" decay is measured to be 7.2x10^-6. This last decay proceedsthrough a second-class current and is expected to be forbidden in the limit ofisospin symmetry.

Measurement of the tau -> 3pi eta nu Branching Fraction and a Search for a Second-Class Current in the tau -> eta'(958) pi nu Decay