Consistent truncations around half-maximal AdS$_5$ vacua of 11-dimensional supergravity

We use exceptional field theory to systematically study all possible consistent truncations around half-maximal AdS$_5$ vacua of M-theory. We show that only truncations with at most three vector multiplets are consistent. The possible gaugings are SU(2)$\times$U(1) and ISO(3)$\times$U(1) where, in the first case, the U(1) factor can have different embeddings inside the global symmetry group SO$(5,n)$, where $n \leq 3$ equals the number of vector multiplets. We show that the ISO(3)$\times$U(1) truncation only exists around vacua that are locally an $S^4$ fibration over a Riemann surface, and argue that this is the only consistent truncation with vector multiplets around these vacua that can be constructed by going through the 7-dimensional SO(5) gauged supergravity obtained after an $S^4$ reduction. The consistent truncations with SU(2)$\times$U(1) gaugings with vector multiplets exist only if certain conditions are satisfied, which we derive.