Solve the following SDE

Question

I have the SDE:

$$X_t=X_0+\int_0^t \sqrt{1-X_s^2}dW_s-\frac{1}{2}\int_0^t X_s ds$$

I get no strong solutions but a weak solution (only one I was able to find):

$$X_t=\text{cos}(W_t),$$

I wonder primarily two questions:

1. Are there other weak solutions that distributionally differ?
2. Is there any solution that works for all real $X_0$ (the above solution requires $X_0$ is 1).