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Let $L$ be a positive definite (non-classic) ternary $\z$-lattice and let $p$be a prime such that a $\frac 12\z_p$-modular component of $L_p$ is nonzeroisotropic and $4\cdot dL$ is not divisible by $p$. For a nonnegative integer$m$, let $\mathcal G_{L,p}(m)$ be the genus with discriminant $p^m\cdot dL$ onthe quadratic space $L^{p^m}\otimes \q$ such that for each lattice $T \in\mathcal G_{L,p}(m)$, a $\frac 12\z_p$-modular component of $T_p$ is nonzeroisotropic, and $T_q$ is isometric to $(L^{p^m})_q$ for any prime $q$ differentfrom $p$. Let $r(n,M)$ be the number of representations of an integer $n$ by a$\z$-lattice $M$. In this article, we show that if $m \le 2$ and $n$ isdivisible by $p$ only when $m=2$, then for any $T \in \mathcal G_{L,p}(m)$,$r(n,T)$ can be written as a linear summation of $r(pn,S_i)$ and $r(p^3n,S_i)$for $S_i \in \mathcal G_{L,p}(m+1)$ with an extra term in some special case. Weprovide a simple criterion on when the extra term is necessary, and we computethe extra term explicitly. We also give a recursive relation to compute$r(n,T)$, for any $T \in \mathcal G_{L,p}(m)$, by using the number ofrepresentations of some integers by lattices in $\mathcal G_{L,p}(m+1)$ for anarbitrary integer $m$.

A generalization of Watson transformation and representations of ternary quadratic forms