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Teleparallel geometry utilizes Weitzenb\"ock connection which has nontrivialtorsion but no curvature and does not directly follow from the metric likeLevi-Civita connection. In extended teleparallel theories, for instance in$f(T)$ or scalar-torsion gravity, the connection must obey its antisymmetricfield equations. So far only a few analytic solutions were known. In this notewe solve the $f(T,\phi)$ gravity antisymmetric vacuum field equations for ageneric rotating tetrad ansatz in Weyl canonical coordinates, and find thecorresponding spin connection coefficients. By a coordinate transformation wepresent the solution also in Boyer-Lindquist coordinates, often used to studyrotating solutions in general relativity. The result hints for the existence ofanother branch of rotating solutions besides the Kerr family in extendedteleparallel gravities.

Flat connection for rotating vacuum spacetimes in extended teleparallel gravity theories