We report experimentally and theoretically a new state in which injection locking of the parallel polarization and simultaneous excitation of the orthogonal polarization in a single-transverse mode VCSEL are obtained. We find very simple analytical expressions for the power of both linear polarizations. A simple nonlinear dependence between the power of both linear polarizations and the frequency detuning is found. Also, the total power emitted by the VCSEL is constant and it is independent of the injected power and of the frequency detuning. The stability maps in the injected power-frequency detuning plane identifying this state are performed for different values of bias currents with good agreement between the experimental and calculated maps. Moreover, a simple analytical expression that describes the boundaries of these maps for negative and large frequency detunings provides a simple method to extract the linear dichroism of the laser.