Yes you can use exact logistic regression when sample size is small. But I do not know if there is a fast rule to find the sample size or number of variables.
How many observations and variables (to be included in the model) do you have?
Event number less than twenty in my study. This is the problem for conventional logistic regression analysis. Practically, if you have twenty cases with positive result for tested condition, you can include only two variables to model of logistic regression. So, my questions: Can I solve this problem with exact logistic regression ? How to perform this analysis?
What are the exact number of events and nonevents in your study?
You can use exact logistic regression.
Maybe conventional logistic regression with Firth method would also fit to your case. You can do both on R,SAS or Stata. It is just like regular logistic regression.
I have not heard the practical rule you are stating. Any references? I think you are using 10 observations per variable rule(!) to find the maximum number of predictors.