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The (second-order) tensor equation AX + XA = ~(A, H) is studied for certain isotropic functions ,I,(A, I-I) which are linear in H. Qualitative properties of the solution X and relations between the solutions for various forms of'I' are established for an inner product space of arbitrary dimension.These results, together with Rivlin's identities for tensor polynomials in two variables, are applied in three dimensions to obtain new explicit formulas for X in direct tensor notation as well as new derivations of previously known formulas. Several applications to the kinematics of continua are considered.

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