I. Modal auxiliaries and periphrastic do have defective paradigms:
(a) modal auxiliaries have no:
(i) infinitive, e.g. * to can
(ii) imperative, e.g. * Can!
(iii) participle, e.g. * is canning/has could
(iv) agreement marking on the 3rd person singular form, e.g. * she cans
(b) periphrastic do has
(i) no infinitive, * He tried to do not sleep
(ii) no participle, * is doing not sleep
(iii) an imperative, Do (not) leave!
(iv) agreement marking on the 3rd person singular form, Does he sleep?
(BUT: main verb do does have infinitival and participial forms, He has done so. He tried to do so.)
II. Auxiliary have and be have complete morphological paradigms.
Modal auxiliaries, auxiliary have and be, and periphrastic do form three different morphological paradigms.