Explicit formulas concerning eigenvectors of weakly non-unitary matrices

We investigate eigenvector statistics of the Truncated Unitary ensemble TUE(N,M) in weakly non-unitary case M=1, that is when only one row and column are removed. provide an explicit description generalized overlaps as deterministic functions eigenvalues, well a method to derive exact formula for expectation diagonal (or squared eigenvalue condition numbers), conditionally on eigenvalue. This complements recent results obtained opposite regime M≥N, suggesting possible extensions all values M.