In this article, the authors consider an elliptic curve $E$ with complex multiplication (CM), given by a short Weierstrass equation with coefficients in a number field $k$, and a curve $\Cal{C} \subset E^N$ that is transverse, i.e., not contained in any translate of a proper algebraic subgroup of $E^N$ (defined over an algebraic closure of $k$). \par The article's main result is an explicit upper bound for the (Néron-Tate) height of any point on $\Cal{C}$ that is contained in a proper algebraic subgroup. The bound depends on $N$, the degree and the height of $\Cal{C}$ embedded in $\Bbb{P}^{3^N-1}$ via the Segre embedding, the discriminant of the fraction field $K$ of the geometric endomorphism ring $\roman{End}(E)$ of $E$, the conductor of $\roman{End}(E)$, and the maximal difference between the canonical and the naive height of a point on $E$ (for which an explicit upper bound is provided in the article). \par Qualitatively, and also for $E$ without CM, this result is due to E. Viada [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) {\bf 2} (2003), no.~1, 47--75; MR1990974] and has its roots in an analogous result of E. Bombieri, D.~W. Masser and U.~M. Zannier for powers of the multiplicative group in [Internat. Math. Res. Notices {\bf 1999}, no.~20, 1119--1140; MR1728021]. In [Trans. Amer. Math. Soc. {\bf 369} (2017), no.~9, 6465--6491; MR3660229], the authors of the article under review together with S. Checcoli obtained an effective upper bound for the height of a point on $\Cal{C}$ that is contained in a proper algebraic subgroup of dimension at most one, also for $E$ without CM and explicit in that case. Important ingredients of the proof of the main result of the article under review are the arithmetic Bézout theorem as well as Minkowski's first and second theorems. \par The obtained upper bound for the height is also an upper bound for the height of any point in $\Cal{C}(k)$ as soon as the $\roman{End}(E)$-module generated by $E(k)$ has rank $MR4010563] to explicitly determine $\Cal{C}(k)$ in some cases. For instance, they determine all solutions in $\Bbb{Q}(\sqrt{-3})$ to the system of equations $y_1^2 = x_1^3 + 2$, $y_2^2 = x_2^3 + 2$, $x_1^n = y_2$ for every choice of ${n \geq 1}$.