GW-invariants and the reduced genus $1$ GW-invariants constructed in [@g1comp2] for all symplectic manifolds.\
By the proof of [@McS Theorem 3.1.5], for a generic almost complex structure $J$ on $X$ all moduli spaces $\M_{h,\eset}^*(X,\be';J)$ are smooth and of the expected dimension, $2\lr{c_1(TX),\be'}$. In particular, &lt;0 \_[h,S]{}\^\*(X,’;J),\_[h,S]{}(X,’;J)=.By a similar argument, for a generic $J$ on $X$ the evaluation maps $$\ev_1,\ev_2\!:  \M_{g,\{1,2\}}^*(X,\be;J)\lra X$$ are transverse, while the bundle section $$\M_{g,\{1\}}^*(X,\be;J)\lra L_1^*\!\otimes\!\ev_1^*TX, \qquad
[u]\lra d_{z_1(u)}u\,,$$ where $L_1\!\lra\!\M_{g,\{1\}}^*(X,\be;J)$ is the universal tangent line bundle at the marked point and $z_1(u)\!\in\!\Si_u$ is the marked point of $u$, is transverse to the zero set. Thus, $$\M_{g,S}^{sing}(X,\be;J)\equiv
\big\{[u]\!\in\!\M_{g,S}^*(X,\be;J)\!:~
u~\textnormal{is not an embedding}\big\}$$ is the image of a smooth map from a smooth manifold of (real) dimension two less than the dimension of $\M_{g,S}^*(X,\be;J)$. It follows that for a generic choice of pseudocycle representatives $f_j\!:M_j\!\lra\!X$ for $\ka_j$, $\M_{g,\f}^*(X,\be;J)$ is a $0$-dimensional oriented sub-manifold of $$\big(\M_{g,S}^*(X,\be;J)-\M_{g,S}^{sing}(X,\be;J)\big)
\times\prod_{j\in S}\!M_j.$$\
We next show that $\M_{g,\f}^*(X,\be;J)$ is a finite set. If not, there is a sequence $([u_r],(w_{r,j})_{j\in S})$ in $\M_{g,\f}^*(X,\be;J)$ converging to some $$\big([u],(w_j)_{j\in S}\big)\in 
\ov\M_{g,\f}(X,\be;J)- \M_{g,\f}^*(X,\be;J).$$ The image of $u$ is a connected $J$-holomorphic curve in $X$ of genus $h\!\le\!g$ with $k\!\ge\!1$ irreducible components of degrees $\be_1,\ldots,\be_k\!\in\!H_2(X;\Z)$ such that $$d_1\be_1+\ldots+d_k\be_k=\be \qquad\hbox{for 