An analog of Lie’s second fundamental theorem for infinite dimensional Lie groups is: Let \(G\) be such a group, \(\mathcal G\) the Lie algebra of \(G\) and \(\mathcal H\) a closed subalgebra of \(\mathcal G\) then there exists a Lie subgroup \(H\) of \(G\) such that \({\mathcal H}\) is the Lie algebra of \(H\). In general this statement is known to be false.
In this paper the author defines a class of groups \(G\) and a class of subalgebras \(\mathcal H\) of \(\mathcal G\) and proves the statement is true for these classes. Some particular cases: (a) \(G\) is a Banach Lie group, \(\mathcal H\) having a closed complement in \(\mathcal G\); (b) \(G = \text{Diff }M\) with \(M\) being a compact manifold without boundary, \(\mathcal H\) consists of the divergence free vector fields; (c) \(G\) as in (b), \(\mathcal H\) consists of the locally Hamiltonian vector fields.