Isomorphic limit ultrapowers for infinitary logic

The logic $${\mathbb{L}}_\theta ^1$$ was introduced in [She12]; it is the maximal below $${{\mathbb{L}}_{\theta, \theta}}$$ which a well ordering not definable. We investigate for θ compact cardinal. prove that satisfies several parallels of classical theorems on first order logic, strengthening thesis natural logic. In particular, two models are -equivalent iff some ω-sequence θ-complete ultrafilters, iterated ultrapowers by those isomorphic. Also strong limit λ>θ cofinality $${\aleph _0}$$ , every complete -theory has so-called special model cardinality λ, parallel saturated. For theory T and singular cardinal λ. Using “special” our context justified by: unique (fixing λ), all reducts too, so we have another proof interpolation this case.