Embedding 5-cycle systems into pentagon triple systems

We show that the spectrum for pentagon triple systems is the set of all n ≡ 1 , 15 , 21 or 25 ( mod 30 ) . We then construct a 5-cycle system of order 10 n + 1 which can be embedded in a pentagon triple system of order 30 n + 1 and also construct a 5-cycle system of order 10 n + 5 which can be embedded in a pentagon triple system of order 30 n + 15 , with the possible exception of embedding a 5-cycle system of order 21 in a pentagon triple system of order 61.