I read about convertible engines. I want to know if there is any engine existing to meet up hover and cruise together.
The closest one is the P&W F-135, used in the F-35B VTOL-capable fighter. However, it switches between turbofan and turbofan+turboshaft modes. The fan's still running, but its flow is redirected down through thrust vectoring for hovering.
Here it is shown driving a lift fan, which includes the gearbox. A similar but larger gearbox can be used to drive a helicopter rotor, if necessary.
This particular engine is too large for a practical-sized helicopter. But it might be possible to convert smaller engines in a similar way.