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In this paper we study a special subclass of real solvable Lie algebrashaving small dimensional or small codimensional derived ideal. It is well-knownthat the derived ideal of any Heisenberg Lie algebra is 1-dimensional and thederived ideal of the 4-dimensional real Diamond algebra is 1-codimensional.Moreover, all the coadjoint orbits of any Heisenberg Lie group as well as4-dimensional real Diamond group are orbits of dimension zero or maximaldimension. In general, a (finite dimensional) real solvable Lie group is calledan $MD$-group if its coadjoint orbits are zero-dimensional or maximaldimensional. The Lie algebra of an $MD$-group is called an $MD$-algebra and theclass of all $MD$-algebras is called $MD$-class. Simulating the mentioned abovecharacteristic of Heisenberg Lie algebras and 4-dimensional real Diamondalgebra, we give a complete classification of $MD$-algebras having1-dimensional or 1-codimensional derived ideals.

Classification of Real Solvable Lie Algebras Whose Simply Connected Lie Groups Have Only Zero or Maximal Dimensional Coadjoint Orbits