Lagrangian translating solitons and special lagrangians in Cᵐ with symmetries

We construct novel families of exact immersed and embedded Lagrangian translating solitons and special Lagrangian submanifolds in Cᵐ C m that are invariant under the action of various admissible compact subgroups G ≤ \,SU\,(m-1) G ≤ SU ( m - 1 ) with cohomogeneity-two. These examples are obtained via an Ansatz generalising a construction of Castro–Lerma in C² 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 \,SU\,(m) Cᵐ SU ( m ) ⋉ C m , including cohomogeneity-one examples.