  Lightshift of a state due to the applied laser in an atomic system vanishes
at the tune-out wavelengths ($\lambda_T$s). Similarly, differential light shift
of a transition vanishes at the magic wavelengths ($\lambda_{magic}$s). In many
of the earlier studies, values of the electric dipole (E1) matrix elements were
inferred precisely by combining measurements of $\lambda_{magic}$ with the
calculated their values. Similarly, the $\lambda_T$ values of an atomic state
can be used to infer the E1 matrix element as it involves dynamic electric
dipole ($\alpha$) values of only one state whereas the $\lambda_{magic}$ values
are dealt with $\alpha$ values of two states. However, both the $\lambda_T$ and
$\lambda_{magic}$ values depend on angular momenta and their magnetic
components ($M$) of states. Here, we report the $\lambda_T$ and
$\lambda_{magic}$ values of many $S_{1/2}$ and $D_{3/2,5/2}$ states, and
transitions among these states of the Mg$^{+}$, Ca$^{+}$, Sr$^{+}$ and Ba$^{+}$
ions that are independent of $M$- values. Measuring these wavelengths in a
special set-up as discussed in the paper, it could be possible to infer a large
number of E1 matrix elements of the above ions accurately.
