A \((v,\rho _2,k,l)\) balanced ternary design (BTD) is a \(v\)-set with multi-set blocks of size \(k\), where each element occurs at most two times in a block, each \(\{x,x\}\) is in \(\rho _2\) blocks and each \(\{x,y\}\), \(x \neq y\), is in \(\lambda \) blocks. By the fine structure of a BTD one understands a sequence \((c_1,c_2,\dots ,c_k)\), where \(c_i\) is the number of blocks with \(i\) repetitions. The paper is concerned with the fine structure in the case when \(k = \lambda = 3\) and \(\rho _2 \in \{1,2\}\). In this case the fine structure is determined by a pair \((t,s)\), where \(c_2 = t\) and \(s + c_3 = v(s-2\rho _2 - 1)/6\). All possible values \((t,s)\) are found for every relevant \(v\). The proof uses counting arguments, various designs found earlier, and constructions that involve holes and group divisible designs.