We construct novel families of exact immersed and embedded Lagrangian translating solitons and special Lagrangian submanifolds in \(\mathbb {C}^m\) that are invariant under the action of various admissible compact subgroups \(G \le {{\,\textrm{SU}\,}}(m-1)\) with cohomogeneity-two. These examples are obtained via an Ansatz generalising a construction of Castro–Lerma in \(\mathbb {C}^2\) . We give explicit examples of admissible group actions, including a full classification for G simple. We also describe novel Lagrangian translators symmetric with respect to non-compact subgroups of the affine special unitary group \({{\,\textrm{SU}\,}}(m)\ltimes \mathbb {C}^m\) , including cohomogeneity-one examples.
