One of the questions of the longest open standing in the area of Lindenmayer-systems is the decidability of the equivalence problem for deterministic, informationless L-systems (DOL-Systems). This and some related equivalence-problems (equivalence with respect to the set and the sequence of generated words, Parikh-vectors and word-lengths) are investigated. Some of these related problems are shown to be recursively solvable, and the implications of these results on the main open problem mentioned above are discussed. (The paper has been accepted for publication in Information and Control).