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We define a group of extended non-Abelian gauge transformations for tensor gauge fields. On this group one can define generalized field strength tensors, which are transforming homogeneously with respect to the extended gauge transformations. The generalized field strength tensors allow to construct two infinite series of gauge invariant quadratic forms. The invariant Lagrangian is a linear sum of these forms and describes interaction of tensor gauge fields of arbitrarily large integer spins 1,2,... . It does not contain higher derivatives of the tensor gauge fields, and all interactions take place through three- and four-particle exchanges with dimensionless coupling constant. The first term in this sum is the Yang-Mills Lagrangian and the whole construction generalizes the Yang-Mills theory. The invariance with respect to the extended gauge transformations does not fix some of the coefficients in the Lagrangian. There is a freedom to vary them without breaking the extended gauge symmetry. We demonstrate that by an appropriate tuning of these coupling constants one can achieve an enhancement of the extended gauge symmetry. This leads to highly symmetric equations. We present the explicit form of the free equations for the rank-2 and rank-3 gauge fields. Their relation to the Schwinger free equation for Abelian rank-3 gauge field is discussed.

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